by Donald W. Brooks
Determining
Static Thrust
February 2008 33
Is your model’s power plant
giving you all it’s got?
I HANDED THE electric-powered GWS
Zero to Ken Marler, a long-time modeler,
and pushed the throttle to full. The motor
wound up instantly and Ken tilted the
Zero’s nose up at an angle to get an idea
of the thrust that was being produced.
“Yep,” he said with the utmost
confidence. “It’ll fly!”
And it did.
Not all modelers have such a finely
developed feel for models’ flyability.
With Scale aircraft, which tend to be on
the heavy side in terms of wing loading,
we may want more assurance than a
nose-up check. This may be particularly
true if we are competing at a flying field
that is at a different elevation from our
home field. Or the Scale model may be
too large to handle that way.
Once we know how much thrust is
required to fly a Scale model, we may
want to make sure that much thrust is
available before the next flight, which
may be conducted under different
atmospheric conditions (temperature and
air pressure). With 1/4-scale 3-D models,
we may want to assure ourselves that the
available static thrust is sufficient for
hovering or knife-edge maneuvers.
We need a way to determine static
thrust quickly and easily. You can attach
your fish scale to a rope around the
model’s tail and run the engine or motor
at full power to measure the thrust. But
determining thrust is harder if your
model does not have landing gear.
What if you could use a tachometer
reading and a four-function calculator
rather than the fish scale to determine
static thrust? Wouldn’t that be much more
convenient?
Wouldn’t it be nice to monitor staticthrust
changes as the temperature changes
and calculate thrust when you go to a
different flying field with a different
elevation? Air density is nonconstant; it
changes with temperature and with air
pressure.
I will describe an easy way to calibrate
propellers and determine the static thrust
a propeller produces from a tachometer
reading. You calibrate the propeller with
a model that has a tail wheel and then use
the propeller on any model in your fleet,
and you will still be able to calculate the
thrust that is being produced.
This article will teach you how to use
simple equations to obtain propellerthrust
coefficients and calculate static
thrust.
Sometimes it would be handy to know
the static thrust a model’s propulsion
system produces. For instance, if you
know from experience how much static
thrust is required for your model to fly
and you want to know if it would be able
to accelerate and take off, you need to
know the thrust produced. See the sidebar
about Russ Crawford’s PT-3.
If your thing is hovering or knife-edge
flight, you might even want to know
whether or not your aircraft has enough
static thrust to perform vertical
maneuvers. Barry Cazier’s Yak-54 with
its 23 x 8 propeller performs differently
in the cool morning air than it does in the
warmer mid- to late-afternoon air of a hot
day. It would be handy to determine the
Yak-54’s static thrust before an afternoon
takeoff to ensure that its performance
will be as desired.
Let’s explore takeoff and vertical
maneuvering. An old rule of thumb
floating around is that if a model’s static
thrust is equal to or greater than one-third
its weight, it will take off from a shortcut
grass field. One distributor even
published this guideline in its catalog.
I believe the rule applied to aircraft
with flat-bottom wings. A model with a
symmetrical airfoil may require more
thrust for takeoff. This rule of thumb has
worked well for me because most of my
airplanes have flat-bottomed airfoils.
I have an 84-ounce, electric-powered
Stick 40, for which the power system
produces a static thrust of 41 ounces.
That is more than 28, which is one-third
of 84. The model takes off quickly and
flies well. It will loop from level flight
but will not do vertical maneuvers. Using
the static-thrust coefficient and a
tachometer reading to determine the
available thrust can tell you if your
airplane has sufficient thrust for takeoff.
02sig2.QXD 12/20/07 9:37 AM Page 3334 MODEL AVIATION
Photos by the author Flight photos by Ron Goodwin and Barry Cazier
Left: Russ Crawford uses a Tower
Hobbies tachometer to determine his
PT-3’s rpm measurements.
Above: After testing, thrust for the PT-3
went from 25 to 28 ounces—a 12% gain.
Three of these
four instruments
are essential for
determining the
thrust coefficient.
Figure 1 can be
used in place of the
hiking barometer if
need be. The
barometer is
optional.
A 15-pound fish
scale is adequate
for models up to
120 size or 11-
pound 3-D aircraft.
Figure 1: Variation of Air Pressure
with Flying Field Elevation
(U.S. Standard Atmosphere - 1962)
20
25
30
0 2000 4000 6000 8000 10000
Elevation
(feet above sea level)
Air Pressure
(Inches of Hg)
When powering a new model, one
approach is to ensure that a certain
number of watts per pound or
horsepower per pound is available from
the power system. I will not discuss that
here. However, once you have selected
the power system, you can use the
propeller static-thrust coefficient and a
tachometer reading to determine the
thrust that is available from your
model’s power system to ensure that the
performance is adequate.
The ability to accelerate to flight
speeds, hover a model, or fly knife-edge
maneuvers depends on propeller thrust.
An extremely powerful motor or engine
that is poorly matched to a propeller
may produce less-than-desired thrust.
The Yak-54 Barry Cazier flies can do
vertical maneuvers. When we tested
Barry’s power system with its NX 23 x
8 propeller, we determined the thrust
coefficient to be 0.054. It produced 18
pounds of thrust to pull the 16.5-pound
model vertically straight up. Once we
had measured the static-thrust
02sig2.QXD 12/20/07 9:37 AM Page 34February 2008 35
Why Concern Yourself With Static Thrust?
1) Some Scale models, such as Russ Crawford’s PT-3 shown in one of
the photographs, are flying close to the limit for available thrust and
wing loading. The model may perform well at a 4,700-foot elevation in
the cool morning air but not be able to take off in the afternoon when
the air is 30° warmer.
Being able to determine the thrust with a tachometer reading helps.
Russ knows that if the PT-3’s static thrust drops below 25 ounces, he
should ground the model and come back to fly another day.
2) A Scale model may fly well at sea level outside Pensacola, Florida.
Take that same airplane to Reno, Nevada, to fly in a competition, and it
may not fly or may not fly as well because of the lower air density
caused by the reduction in air pressure. You may need to change the
propeller to get improved thrust performance at the higher elevation.
If you know the thrust coefficient and have a tachometer, you can
determine the available thrust and make necessary adjustments. Going
from sea level to 4,500 feet along with a 30° increase in temperature
may lower the air density by 20%. The decreased density affects wing
lift, propeller thrust, and engine performance.
The cumulative effects may give your model only 50% of the
performance it had at sea level. If this is your situation, you need to know
how much thrust you can depend on before the first flight at altitude.
3) Barry Cazier’s Yak-54 performs differently in the morning than in
the afternoon because of the temperature effect on air density. He likes
to fly vertical maneuvers such as the Hover, the Waterfall, and knifeedge
flight.
When we measured the thrust produced, it was approximately 18
pounds for a 16.5-pound aircraft. Barry had roughly a 9% excess of
thrust. Once he has the thrust coefficient, he can tach the propeller at
full throttle and determine quickly how much thrust he has to maneuver
the model.
4) A CL flier looks for consistent lap times for performing aerobatics
with a model. Changing the flying-field elevation or air temperatures
may necessitate a propeller change.
Lee Powell, a modeler in the local flying club, lives in Idaho Falls,
Idaho (elevation 4,740 feet), and competes in Oregon (elevation roughly
400 feet). He flies with a shallower-pitched propeller on his CL model at
the lower elevation to adjust the engine rpm, propeller thrust, and pitch
speed for consistent lap times. MA
—Donald W. Brooks
coefficient, Barry had the option of
using that propeller-thrust coefficient
and a tachometer reading to determine
the available thrust.
Basic Static-Thrust Equation: I use a
basic equation for propeller static thrust
that was provided on page 447 of an
engineering text by John A. Roberson
and Clayton T. Crowe, titled
Engineering Fluid Dynamics. The
authors were professors at Washington
State University, Pullman.
In that textbook they give the staticthrust
equation for aircraft propellers. I
experimented with model propellers
using that equation and verified that it
works just as well for 1:6-scale aircraft
propellers as it does for the 1:1
versions. And the equation works as
well now as it did in 1975; its basic
physics have not changed within the
scale range or with time.
The equation gives us a tool to
determine a propeller’s static thrust if
we know its thrust coefficient, the air
density, the operating propeller’s rpm,
and the propeller’s diameter. It is
shown in a sidebar as Equation 1.
The only hard things to obtain in
that equation are the propeller
coefficient and the air density. The
propeller-thrust coefficient must be
measured or obtained from a cataloged
source, such as my book Prop Talk.
You can calculate the air density if
you know the air pressure and
temperature. There are three ways to
get the air pressure:
• Call the local weather station and ask
for the local barometric air pressure.
• Read the value on a hiking barometer
such as the one Sun Corporation
markets.
• Use Figure 1 to estimate the air
pressure.
Figure 1 shows air pressure vs.
elevation for the U.S. Standard
Atmosphere of 1962. That figure will
also give you a working estimate for
locations where you might know the
elevation but do not have a barometer
or weather person to contact.
You can get the temperature from
your thermometer. You get the rpm
using a tachometer with your engine or
motor running at full power. Measure
the propeller’s diameter or read it off
the propeller.
I adapted Equation 1 to get it in
units that made it easy to use. The
result is Equation 2. Another form of
Equation 2 that may be easier to use on
a simple four-function calculator is
Equation 3. For it I simply eliminated
Barry Cazier’s Yak-54 flies a knife-edge pass. This maneuver
requires a significant amount of propeller thrust.
02sig2.QXD 12/20/07 9:39 AM Page 3536 MODEL AVIATION
Static-Thrust Equations
Equation 1: General Equation for Static Thrust
f = Ct r n2 D4 where
Ct = thrust coefficient for the propeller
r = density of air in which the propeller is working
n = propeller rpm
D = propeller diameter
Equation 2: Calculation of Static Thrust in Ounces
Ct d n2 D4
f = —————————— where
1,000
f = thrust in ounces, a force
d = air density in grams per liter (The word processor I had for
writing Prop Talk could not use Greek letters.)
Ct = the thrust coefficient, a dimensionless coefficient
n = propeller rpm
D = propeller diameter in meters (inches/39.37)
(The factor in the denominator is actually 1002.3 and is a constant
that contains all the unit conversions. I rounded this off to 1,000 for
ease of use in calculations.)
Another form of Equation 2 that may be easier to use on a simple
four-function calculator is:
Equation 3: Calculation of Static Thrust in Ounces
f = 4.1586 Ct d n12 D14 where
f, Ct, d = the same as in Equation 2
n1 = rpm divided by 1,000
D1 = diameter in inches/10
Equation 4: Calculation of Air Density
(11.79) (P)
d = ——————————— where
T
d = air density in grams per liter
P = local air pressure in inches of Hg
T = temperature in degrees Kelvin (Measure the temperature in
degrees Fahrenheit and calculate this temperature from Equation 5.)
Equation 5: Calculation of Temperature in Degrees Kelvin—T deg
K
T deg K = (T deg F – 32) x (5/9) + 273.16
Equation 6: Calculation of Static-Thrust Coefficient
f
Ct = ———————————
4.1586 d n12 D14
—Donald W. Brooks
the conversion to meters and divided
the diameter by 10 and the rpm by
1,000. You can do these preliminary
calculations in your head.
Let’s assume you have measured
your propeller’s rpm and diameter.
Now you want the air density to put
into the equation. The method for
calculating air density is Equation 4.
Once you know the air density, rpm
at full power, and propeller diameter,
you need only the static thrust to
calculate the static-thrust coefficient. If
you have a fish scale, you can hook it
between an anchor point and the
model’s tail. Then, using the propeller
for which you want the thrust
coefficient and running the engine or
motor at full power, measure the static
thrust.
Knowing that number, air density,
rpm, and propeller diameter, you can
use Equation 6 to calculate the
propeller-thrust coefficient. Once you
have the propeller coefficient, you can
easily calculate the thrust with that
propeller on any aircraft and at any
time and location, i.e., temperature and
air pressure (elevation effects), you
want using a tachometer reading and
Equation 3.
What are you waiting for? Find your
four-function calculator, get your
tachometer out of storage, round up a
thermometer and a De-Liar fish scale,
and call your local National Weather
Service person for the local barometric
air pressure or use Figure 1 to estimate
air pressure in your location.
Calibrate some propellers by
measuring the thrust under recorded
conditions. Mark each one you calibrate
with the measured thrust coefficient.
Once calibrated, make a full-power
tachometer measurement of rpm and
calculate your model’s static thrust.
Yes, it takes a little preparation.
You have to calibrate each propeller
you want to use in this process or you
must get the thrust coefficient from a
cataloged source. If you do not already
have the thrust coefficient for a
propeller, get out your fish scale, start
the propulsion system, make the
measurements, and calculate the thrust
coefficient.
As long as you use that propeller
and it is undamaged, the thrust
coefficient is good. You are now armed
with the knowledge to determine and
use the static-thrust coefficients on your
own.
Whether you get the thrust
coefficients from a cataloged source or
determine your own, get into the habit
02sig2.QXD 12/20/07 9:39 AM Page 36February 2008 37
Measuring Propeller Static-Thrust Coefficient
Let’s look at an example. With a calibrated thrust-measuring device, I
determined that a Zinger 14 x 10 propeller produces 41 ounces of thrust.
This propeller was operating at 4,500 rpm, the air temperature was 74
degrees Fahrenheit, and the barometric air pressure was 24.95 inches Hg.
What is this propeller’s thrust coefficient?
First, calculate D1 and n1.
D1 = 14 inches/10 = 1.4
n1 = 4,500 rpm/1,000 = 4.5
Second, calculate the temperature in Kelvin using Equation 5.
T (deg Kelvin) = (74 deg F – 32 deg F) (5/9) + 273.16
T = 296.5 deg Kelvin
Third, calculate the air density using Equation 4.
(11.79) (24.95)
d = ————————————— grams per liter
296.5
d = 0.9921 grams per liter
Fourth, calculate the static-thrust coefficient using Equation 6.
(41)
Ct = ——————————————————
4.1586 (0.9921) (4.5)2 (1.4)4
Ct = 0.128
We have determined that this Zinger 14 x 10 propeller’s thrust coefficient
is 0.128. We use a black permanent marker to write this figure on the
backside of the propeller. Now we can use Equation 3 to calculate the thrust
for different conditions of rpm or air density. MA
—Donald W. Brooks
of calculating your system’s static
thrust when preparing for flight. This
is particularly important if you are
preparing to fly a high-value model in
an unfamiliar location. Make sure you
obtain the current barometric air
pressure or use the flying-field
elevation to get an air-pressure
estimate from Figure 1.
If you know how much thrust is
required for your model to fly your
kind of mission, you can determine
the static thrust that is produced and
ensure that the thrust is adequate
before takeoff. It’s better to have an
aborted flight than a major repair or
total destruction of a model. You will
have greater confidence in your
aircraft’s flyability.
Great flying and fly safely! MA
Donald W. Brooks
[email protected]
Sources:
Altimeter (item 203):
Sun Company, Inc.
www.suncompany.net
(800) 441-0132
Engineering Fluid Dynamics (ISBN:
0-395-18607-2), 1975:
Houghton Mifflin Company
www.hmco.com
Prop Talk (ISBN 0-9657014-0-9),
1997:
ARPI Publishing
900 Bower Dr.
Idaho Falls ID 83404
Normark Weigh-In electronic digital
15-pound scale:
10395 Yellow Circle Dr.
Minnetonka MN 55343
(952) 933-7060
The author marks the back of his propellers with their pitch values and the
thrust and power coefficients.
OWN A MACHINE SHOP
1-800-476-4849 E xt. MA
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02sig2.QXD 12/20/07 9:40 AM Page 37
Edition: Model Aviation - 2008/02
Page Numbers: 33,34,35,36,37,38
Edition: Model Aviation - 2008/02
Page Numbers: 33,34,35,36,37,38
by Donald W. Brooks
Determining
Static Thrust
February 2008 33
Is your model’s power plant
giving you all it’s got?
I HANDED THE electric-powered GWS
Zero to Ken Marler, a long-time modeler,
and pushed the throttle to full. The motor
wound up instantly and Ken tilted the
Zero’s nose up at an angle to get an idea
of the thrust that was being produced.
“Yep,” he said with the utmost
confidence. “It’ll fly!”
And it did.
Not all modelers have such a finely
developed feel for models’ flyability.
With Scale aircraft, which tend to be on
the heavy side in terms of wing loading,
we may want more assurance than a
nose-up check. This may be particularly
true if we are competing at a flying field
that is at a different elevation from our
home field. Or the Scale model may be
too large to handle that way.
Once we know how much thrust is
required to fly a Scale model, we may
want to make sure that much thrust is
available before the next flight, which
may be conducted under different
atmospheric conditions (temperature and
air pressure). With 1/4-scale 3-D models,
we may want to assure ourselves that the
available static thrust is sufficient for
hovering or knife-edge maneuvers.
We need a way to determine static
thrust quickly and easily. You can attach
your fish scale to a rope around the
model’s tail and run the engine or motor
at full power to measure the thrust. But
determining thrust is harder if your
model does not have landing gear.
What if you could use a tachometer
reading and a four-function calculator
rather than the fish scale to determine
static thrust? Wouldn’t that be much more
convenient?
Wouldn’t it be nice to monitor staticthrust
changes as the temperature changes
and calculate thrust when you go to a
different flying field with a different
elevation? Air density is nonconstant; it
changes with temperature and with air
pressure.
I will describe an easy way to calibrate
propellers and determine the static thrust
a propeller produces from a tachometer
reading. You calibrate the propeller with
a model that has a tail wheel and then use
the propeller on any model in your fleet,
and you will still be able to calculate the
thrust that is being produced.
This article will teach you how to use
simple equations to obtain propellerthrust
coefficients and calculate static
thrust.
Sometimes it would be handy to know
the static thrust a model’s propulsion
system produces. For instance, if you
know from experience how much static
thrust is required for your model to fly
and you want to know if it would be able
to accelerate and take off, you need to
know the thrust produced. See the sidebar
about Russ Crawford’s PT-3.
If your thing is hovering or knife-edge
flight, you might even want to know
whether or not your aircraft has enough
static thrust to perform vertical
maneuvers. Barry Cazier’s Yak-54 with
its 23 x 8 propeller performs differently
in the cool morning air than it does in the
warmer mid- to late-afternoon air of a hot
day. It would be handy to determine the
Yak-54’s static thrust before an afternoon
takeoff to ensure that its performance
will be as desired.
Let’s explore takeoff and vertical
maneuvering. An old rule of thumb
floating around is that if a model’s static
thrust is equal to or greater than one-third
its weight, it will take off from a shortcut
grass field. One distributor even
published this guideline in its catalog.
I believe the rule applied to aircraft
with flat-bottom wings. A model with a
symmetrical airfoil may require more
thrust for takeoff. This rule of thumb has
worked well for me because most of my
airplanes have flat-bottomed airfoils.
I have an 84-ounce, electric-powered
Stick 40, for which the power system
produces a static thrust of 41 ounces.
That is more than 28, which is one-third
of 84. The model takes off quickly and
flies well. It will loop from level flight
but will not do vertical maneuvers. Using
the static-thrust coefficient and a
tachometer reading to determine the
available thrust can tell you if your
airplane has sufficient thrust for takeoff.
02sig2.QXD 12/20/07 9:37 AM Page 3334 MODEL AVIATION
Photos by the author Flight photos by Ron Goodwin and Barry Cazier
Left: Russ Crawford uses a Tower
Hobbies tachometer to determine his
PT-3’s rpm measurements.
Above: After testing, thrust for the PT-3
went from 25 to 28 ounces—a 12% gain.
Three of these
four instruments
are essential for
determining the
thrust coefficient.
Figure 1 can be
used in place of the
hiking barometer if
need be. The
barometer is
optional.
A 15-pound fish
scale is adequate
for models up to
120 size or 11-
pound 3-D aircraft.
Figure 1: Variation of Air Pressure
with Flying Field Elevation
(U.S. Standard Atmosphere - 1962)
20
25
30
0 2000 4000 6000 8000 10000
Elevation
(feet above sea level)
Air Pressure
(Inches of Hg)
When powering a new model, one
approach is to ensure that a certain
number of watts per pound or
horsepower per pound is available from
the power system. I will not discuss that
here. However, once you have selected
the power system, you can use the
propeller static-thrust coefficient and a
tachometer reading to determine the
thrust that is available from your
model’s power system to ensure that the
performance is adequate.
The ability to accelerate to flight
speeds, hover a model, or fly knife-edge
maneuvers depends on propeller thrust.
An extremely powerful motor or engine
that is poorly matched to a propeller
may produce less-than-desired thrust.
The Yak-54 Barry Cazier flies can do
vertical maneuvers. When we tested
Barry’s power system with its NX 23 x
8 propeller, we determined the thrust
coefficient to be 0.054. It produced 18
pounds of thrust to pull the 16.5-pound
model vertically straight up. Once we
had measured the static-thrust
02sig2.QXD 12/20/07 9:37 AM Page 34February 2008 35
Why Concern Yourself With Static Thrust?
1) Some Scale models, such as Russ Crawford’s PT-3 shown in one of
the photographs, are flying close to the limit for available thrust and
wing loading. The model may perform well at a 4,700-foot elevation in
the cool morning air but not be able to take off in the afternoon when
the air is 30° warmer.
Being able to determine the thrust with a tachometer reading helps.
Russ knows that if the PT-3’s static thrust drops below 25 ounces, he
should ground the model and come back to fly another day.
2) A Scale model may fly well at sea level outside Pensacola, Florida.
Take that same airplane to Reno, Nevada, to fly in a competition, and it
may not fly or may not fly as well because of the lower air density
caused by the reduction in air pressure. You may need to change the
propeller to get improved thrust performance at the higher elevation.
If you know the thrust coefficient and have a tachometer, you can
determine the available thrust and make necessary adjustments. Going
from sea level to 4,500 feet along with a 30° increase in temperature
may lower the air density by 20%. The decreased density affects wing
lift, propeller thrust, and engine performance.
The cumulative effects may give your model only 50% of the
performance it had at sea level. If this is your situation, you need to know
how much thrust you can depend on before the first flight at altitude.
3) Barry Cazier’s Yak-54 performs differently in the morning than in
the afternoon because of the temperature effect on air density. He likes
to fly vertical maneuvers such as the Hover, the Waterfall, and knifeedge
flight.
When we measured the thrust produced, it was approximately 18
pounds for a 16.5-pound aircraft. Barry had roughly a 9% excess of
thrust. Once he has the thrust coefficient, he can tach the propeller at
full throttle and determine quickly how much thrust he has to maneuver
the model.
4) A CL flier looks for consistent lap times for performing aerobatics
with a model. Changing the flying-field elevation or air temperatures
may necessitate a propeller change.
Lee Powell, a modeler in the local flying club, lives in Idaho Falls,
Idaho (elevation 4,740 feet), and competes in Oregon (elevation roughly
400 feet). He flies with a shallower-pitched propeller on his CL model at
the lower elevation to adjust the engine rpm, propeller thrust, and pitch
speed for consistent lap times. MA
—Donald W. Brooks
coefficient, Barry had the option of
using that propeller-thrust coefficient
and a tachometer reading to determine
the available thrust.
Basic Static-Thrust Equation: I use a
basic equation for propeller static thrust
that was provided on page 447 of an
engineering text by John A. Roberson
and Clayton T. Crowe, titled
Engineering Fluid Dynamics. The
authors were professors at Washington
State University, Pullman.
In that textbook they give the staticthrust
equation for aircraft propellers. I
experimented with model propellers
using that equation and verified that it
works just as well for 1:6-scale aircraft
propellers as it does for the 1:1
versions. And the equation works as
well now as it did in 1975; its basic
physics have not changed within the
scale range or with time.
The equation gives us a tool to
determine a propeller’s static thrust if
we know its thrust coefficient, the air
density, the operating propeller’s rpm,
and the propeller’s diameter. It is
shown in a sidebar as Equation 1.
The only hard things to obtain in
that equation are the propeller
coefficient and the air density. The
propeller-thrust coefficient must be
measured or obtained from a cataloged
source, such as my book Prop Talk.
You can calculate the air density if
you know the air pressure and
temperature. There are three ways to
get the air pressure:
• Call the local weather station and ask
for the local barometric air pressure.
• Read the value on a hiking barometer
such as the one Sun Corporation
markets.
• Use Figure 1 to estimate the air
pressure.
Figure 1 shows air pressure vs.
elevation for the U.S. Standard
Atmosphere of 1962. That figure will
also give you a working estimate for
locations where you might know the
elevation but do not have a barometer
or weather person to contact.
You can get the temperature from
your thermometer. You get the rpm
using a tachometer with your engine or
motor running at full power. Measure
the propeller’s diameter or read it off
the propeller.
I adapted Equation 1 to get it in
units that made it easy to use. The
result is Equation 2. Another form of
Equation 2 that may be easier to use on
a simple four-function calculator is
Equation 3. For it I simply eliminated
Barry Cazier’s Yak-54 flies a knife-edge pass. This maneuver
requires a significant amount of propeller thrust.
02sig2.QXD 12/20/07 9:39 AM Page 3536 MODEL AVIATION
Static-Thrust Equations
Equation 1: General Equation for Static Thrust
f = Ct r n2 D4 where
Ct = thrust coefficient for the propeller
r = density of air in which the propeller is working
n = propeller rpm
D = propeller diameter
Equation 2: Calculation of Static Thrust in Ounces
Ct d n2 D4
f = —————————— where
1,000
f = thrust in ounces, a force
d = air density in grams per liter (The word processor I had for
writing Prop Talk could not use Greek letters.)
Ct = the thrust coefficient, a dimensionless coefficient
n = propeller rpm
D = propeller diameter in meters (inches/39.37)
(The factor in the denominator is actually 1002.3 and is a constant
that contains all the unit conversions. I rounded this off to 1,000 for
ease of use in calculations.)
Another form of Equation 2 that may be easier to use on a simple
four-function calculator is:
Equation 3: Calculation of Static Thrust in Ounces
f = 4.1586 Ct d n12 D14 where
f, Ct, d = the same as in Equation 2
n1 = rpm divided by 1,000
D1 = diameter in inches/10
Equation 4: Calculation of Air Density
(11.79) (P)
d = ——————————— where
T
d = air density in grams per liter
P = local air pressure in inches of Hg
T = temperature in degrees Kelvin (Measure the temperature in
degrees Fahrenheit and calculate this temperature from Equation 5.)
Equation 5: Calculation of Temperature in Degrees Kelvin—T deg
K
T deg K = (T deg F – 32) x (5/9) + 273.16
Equation 6: Calculation of Static-Thrust Coefficient
f
Ct = ———————————
4.1586 d n12 D14
—Donald W. Brooks
the conversion to meters and divided
the diameter by 10 and the rpm by
1,000. You can do these preliminary
calculations in your head.
Let’s assume you have measured
your propeller’s rpm and diameter.
Now you want the air density to put
into the equation. The method for
calculating air density is Equation 4.
Once you know the air density, rpm
at full power, and propeller diameter,
you need only the static thrust to
calculate the static-thrust coefficient. If
you have a fish scale, you can hook it
between an anchor point and the
model’s tail. Then, using the propeller
for which you want the thrust
coefficient and running the engine or
motor at full power, measure the static
thrust.
Knowing that number, air density,
rpm, and propeller diameter, you can
use Equation 6 to calculate the
propeller-thrust coefficient. Once you
have the propeller coefficient, you can
easily calculate the thrust with that
propeller on any aircraft and at any
time and location, i.e., temperature and
air pressure (elevation effects), you
want using a tachometer reading and
Equation 3.
What are you waiting for? Find your
four-function calculator, get your
tachometer out of storage, round up a
thermometer and a De-Liar fish scale,
and call your local National Weather
Service person for the local barometric
air pressure or use Figure 1 to estimate
air pressure in your location.
Calibrate some propellers by
measuring the thrust under recorded
conditions. Mark each one you calibrate
with the measured thrust coefficient.
Once calibrated, make a full-power
tachometer measurement of rpm and
calculate your model’s static thrust.
Yes, it takes a little preparation.
You have to calibrate each propeller
you want to use in this process or you
must get the thrust coefficient from a
cataloged source. If you do not already
have the thrust coefficient for a
propeller, get out your fish scale, start
the propulsion system, make the
measurements, and calculate the thrust
coefficient.
As long as you use that propeller
and it is undamaged, the thrust
coefficient is good. You are now armed
with the knowledge to determine and
use the static-thrust coefficients on your
own.
Whether you get the thrust
coefficients from a cataloged source or
determine your own, get into the habit
02sig2.QXD 12/20/07 9:39 AM Page 36February 2008 37
Measuring Propeller Static-Thrust Coefficient
Let’s look at an example. With a calibrated thrust-measuring device, I
determined that a Zinger 14 x 10 propeller produces 41 ounces of thrust.
This propeller was operating at 4,500 rpm, the air temperature was 74
degrees Fahrenheit, and the barometric air pressure was 24.95 inches Hg.
What is this propeller’s thrust coefficient?
First, calculate D1 and n1.
D1 = 14 inches/10 = 1.4
n1 = 4,500 rpm/1,000 = 4.5
Second, calculate the temperature in Kelvin using Equation 5.
T (deg Kelvin) = (74 deg F – 32 deg F) (5/9) + 273.16
T = 296.5 deg Kelvin
Third, calculate the air density using Equation 4.
(11.79) (24.95)
d = ————————————— grams per liter
296.5
d = 0.9921 grams per liter
Fourth, calculate the static-thrust coefficient using Equation 6.
(41)
Ct = ——————————————————
4.1586 (0.9921) (4.5)2 (1.4)4
Ct = 0.128
We have determined that this Zinger 14 x 10 propeller’s thrust coefficient
is 0.128. We use a black permanent marker to write this figure on the
backside of the propeller. Now we can use Equation 3 to calculate the thrust
for different conditions of rpm or air density. MA
—Donald W. Brooks
of calculating your system’s static
thrust when preparing for flight. This
is particularly important if you are
preparing to fly a high-value model in
an unfamiliar location. Make sure you
obtain the current barometric air
pressure or use the flying-field
elevation to get an air-pressure
estimate from Figure 1.
If you know how much thrust is
required for your model to fly your
kind of mission, you can determine
the static thrust that is produced and
ensure that the thrust is adequate
before takeoff. It’s better to have an
aborted flight than a major repair or
total destruction of a model. You will
have greater confidence in your
aircraft’s flyability.
Great flying and fly safely! MA
Donald W. Brooks
[email protected]
Sources:
Altimeter (item 203):
Sun Company, Inc.
www.suncompany.net
(800) 441-0132
Engineering Fluid Dynamics (ISBN:
0-395-18607-2), 1975:
Houghton Mifflin Company
www.hmco.com
Prop Talk (ISBN 0-9657014-0-9),
1997:
ARPI Publishing
900 Bower Dr.
Idaho Falls ID 83404
Normark Weigh-In electronic digital
15-pound scale:
10395 Yellow Circle Dr.
Minnetonka MN 55343
(952) 933-7060
The author marks the back of his propellers with their pitch values and the
thrust and power coefficients.
OWN A MACHINE SHOP
1-800-476-4849 E xt. MA
Or Visit us at www.smithy.com
GUARANTEED To pay for itself!
FREE!
Info Kit
FREE!
Info Kit
Call
Today!
“I can fix ‘most anything. I don’t know how I lived without
my Smithy. It paid for itself in no time.”
• Easy to use – No
experience
required.
• Versatile – Fix or
make almost anything.
• Affordable-- 7
models starting
at $999.
• CNC Compatible
Do It Yourself on a Smithy Lathe•Mill•Drill!
02sig2.QXD 12/20/07 9:40 AM Page 37
Edition: Model Aviation - 2008/02
Page Numbers: 33,34,35,36,37,38
by Donald W. Brooks
Determining
Static Thrust
February 2008 33
Is your model’s power plant
giving you all it’s got?
I HANDED THE electric-powered GWS
Zero to Ken Marler, a long-time modeler,
and pushed the throttle to full. The motor
wound up instantly and Ken tilted the
Zero’s nose up at an angle to get an idea
of the thrust that was being produced.
“Yep,” he said with the utmost
confidence. “It’ll fly!”
And it did.
Not all modelers have such a finely
developed feel for models’ flyability.
With Scale aircraft, which tend to be on
the heavy side in terms of wing loading,
we may want more assurance than a
nose-up check. This may be particularly
true if we are competing at a flying field
that is at a different elevation from our
home field. Or the Scale model may be
too large to handle that way.
Once we know how much thrust is
required to fly a Scale model, we may
want to make sure that much thrust is
available before the next flight, which
may be conducted under different
atmospheric conditions (temperature and
air pressure). With 1/4-scale 3-D models,
we may want to assure ourselves that the
available static thrust is sufficient for
hovering or knife-edge maneuvers.
We need a way to determine static
thrust quickly and easily. You can attach
your fish scale to a rope around the
model’s tail and run the engine or motor
at full power to measure the thrust. But
determining thrust is harder if your
model does not have landing gear.
What if you could use a tachometer
reading and a four-function calculator
rather than the fish scale to determine
static thrust? Wouldn’t that be much more
convenient?
Wouldn’t it be nice to monitor staticthrust
changes as the temperature changes
and calculate thrust when you go to a
different flying field with a different
elevation? Air density is nonconstant; it
changes with temperature and with air
pressure.
I will describe an easy way to calibrate
propellers and determine the static thrust
a propeller produces from a tachometer
reading. You calibrate the propeller with
a model that has a tail wheel and then use
the propeller on any model in your fleet,
and you will still be able to calculate the
thrust that is being produced.
This article will teach you how to use
simple equations to obtain propellerthrust
coefficients and calculate static
thrust.
Sometimes it would be handy to know
the static thrust a model’s propulsion
system produces. For instance, if you
know from experience how much static
thrust is required for your model to fly
and you want to know if it would be able
to accelerate and take off, you need to
know the thrust produced. See the sidebar
about Russ Crawford’s PT-3.
If your thing is hovering or knife-edge
flight, you might even want to know
whether or not your aircraft has enough
static thrust to perform vertical
maneuvers. Barry Cazier’s Yak-54 with
its 23 x 8 propeller performs differently
in the cool morning air than it does in the
warmer mid- to late-afternoon air of a hot
day. It would be handy to determine the
Yak-54’s static thrust before an afternoon
takeoff to ensure that its performance
will be as desired.
Let’s explore takeoff and vertical
maneuvering. An old rule of thumb
floating around is that if a model’s static
thrust is equal to or greater than one-third
its weight, it will take off from a shortcut
grass field. One distributor even
published this guideline in its catalog.
I believe the rule applied to aircraft
with flat-bottom wings. A model with a
symmetrical airfoil may require more
thrust for takeoff. This rule of thumb has
worked well for me because most of my
airplanes have flat-bottomed airfoils.
I have an 84-ounce, electric-powered
Stick 40, for which the power system
produces a static thrust of 41 ounces.
That is more than 28, which is one-third
of 84. The model takes off quickly and
flies well. It will loop from level flight
but will not do vertical maneuvers. Using
the static-thrust coefficient and a
tachometer reading to determine the
available thrust can tell you if your
airplane has sufficient thrust for takeoff.
02sig2.QXD 12/20/07 9:37 AM Page 3334 MODEL AVIATION
Photos by the author Flight photos by Ron Goodwin and Barry Cazier
Left: Russ Crawford uses a Tower
Hobbies tachometer to determine his
PT-3’s rpm measurements.
Above: After testing, thrust for the PT-3
went from 25 to 28 ounces—a 12% gain.
Three of these
four instruments
are essential for
determining the
thrust coefficient.
Figure 1 can be
used in place of the
hiking barometer if
need be. The
barometer is
optional.
A 15-pound fish
scale is adequate
for models up to
120 size or 11-
pound 3-D aircraft.
Figure 1: Variation of Air Pressure
with Flying Field Elevation
(U.S. Standard Atmosphere - 1962)
20
25
30
0 2000 4000 6000 8000 10000
Elevation
(feet above sea level)
Air Pressure
(Inches of Hg)
When powering a new model, one
approach is to ensure that a certain
number of watts per pound or
horsepower per pound is available from
the power system. I will not discuss that
here. However, once you have selected
the power system, you can use the
propeller static-thrust coefficient and a
tachometer reading to determine the
thrust that is available from your
model’s power system to ensure that the
performance is adequate.
The ability to accelerate to flight
speeds, hover a model, or fly knife-edge
maneuvers depends on propeller thrust.
An extremely powerful motor or engine
that is poorly matched to a propeller
may produce less-than-desired thrust.
The Yak-54 Barry Cazier flies can do
vertical maneuvers. When we tested
Barry’s power system with its NX 23 x
8 propeller, we determined the thrust
coefficient to be 0.054. It produced 18
pounds of thrust to pull the 16.5-pound
model vertically straight up. Once we
had measured the static-thrust
02sig2.QXD 12/20/07 9:37 AM Page 34February 2008 35
Why Concern Yourself With Static Thrust?
1) Some Scale models, such as Russ Crawford’s PT-3 shown in one of
the photographs, are flying close to the limit for available thrust and
wing loading. The model may perform well at a 4,700-foot elevation in
the cool morning air but not be able to take off in the afternoon when
the air is 30° warmer.
Being able to determine the thrust with a tachometer reading helps.
Russ knows that if the PT-3’s static thrust drops below 25 ounces, he
should ground the model and come back to fly another day.
2) A Scale model may fly well at sea level outside Pensacola, Florida.
Take that same airplane to Reno, Nevada, to fly in a competition, and it
may not fly or may not fly as well because of the lower air density
caused by the reduction in air pressure. You may need to change the
propeller to get improved thrust performance at the higher elevation.
If you know the thrust coefficient and have a tachometer, you can
determine the available thrust and make necessary adjustments. Going
from sea level to 4,500 feet along with a 30° increase in temperature
may lower the air density by 20%. The decreased density affects wing
lift, propeller thrust, and engine performance.
The cumulative effects may give your model only 50% of the
performance it had at sea level. If this is your situation, you need to know
how much thrust you can depend on before the first flight at altitude.
3) Barry Cazier’s Yak-54 performs differently in the morning than in
the afternoon because of the temperature effect on air density. He likes
to fly vertical maneuvers such as the Hover, the Waterfall, and knifeedge
flight.
When we measured the thrust produced, it was approximately 18
pounds for a 16.5-pound aircraft. Barry had roughly a 9% excess of
thrust. Once he has the thrust coefficient, he can tach the propeller at
full throttle and determine quickly how much thrust he has to maneuver
the model.
4) A CL flier looks for consistent lap times for performing aerobatics
with a model. Changing the flying-field elevation or air temperatures
may necessitate a propeller change.
Lee Powell, a modeler in the local flying club, lives in Idaho Falls,
Idaho (elevation 4,740 feet), and competes in Oregon (elevation roughly
400 feet). He flies with a shallower-pitched propeller on his CL model at
the lower elevation to adjust the engine rpm, propeller thrust, and pitch
speed for consistent lap times. MA
—Donald W. Brooks
coefficient, Barry had the option of
using that propeller-thrust coefficient
and a tachometer reading to determine
the available thrust.
Basic Static-Thrust Equation: I use a
basic equation for propeller static thrust
that was provided on page 447 of an
engineering text by John A. Roberson
and Clayton T. Crowe, titled
Engineering Fluid Dynamics. The
authors were professors at Washington
State University, Pullman.
In that textbook they give the staticthrust
equation for aircraft propellers. I
experimented with model propellers
using that equation and verified that it
works just as well for 1:6-scale aircraft
propellers as it does for the 1:1
versions. And the equation works as
well now as it did in 1975; its basic
physics have not changed within the
scale range or with time.
The equation gives us a tool to
determine a propeller’s static thrust if
we know its thrust coefficient, the air
density, the operating propeller’s rpm,
and the propeller’s diameter. It is
shown in a sidebar as Equation 1.
The only hard things to obtain in
that equation are the propeller
coefficient and the air density. The
propeller-thrust coefficient must be
measured or obtained from a cataloged
source, such as my book Prop Talk.
You can calculate the air density if
you know the air pressure and
temperature. There are three ways to
get the air pressure:
• Call the local weather station and ask
for the local barometric air pressure.
• Read the value on a hiking barometer
such as the one Sun Corporation
markets.
• Use Figure 1 to estimate the air
pressure.
Figure 1 shows air pressure vs.
elevation for the U.S. Standard
Atmosphere of 1962. That figure will
also give you a working estimate for
locations where you might know the
elevation but do not have a barometer
or weather person to contact.
You can get the temperature from
your thermometer. You get the rpm
using a tachometer with your engine or
motor running at full power. Measure
the propeller’s diameter or read it off
the propeller.
I adapted Equation 1 to get it in
units that made it easy to use. The
result is Equation 2. Another form of
Equation 2 that may be easier to use on
a simple four-function calculator is
Equation 3. For it I simply eliminated
Barry Cazier’s Yak-54 flies a knife-edge pass. This maneuver
requires a significant amount of propeller thrust.
02sig2.QXD 12/20/07 9:39 AM Page 3536 MODEL AVIATION
Static-Thrust Equations
Equation 1: General Equation for Static Thrust
f = Ct r n2 D4 where
Ct = thrust coefficient for the propeller
r = density of air in which the propeller is working
n = propeller rpm
D = propeller diameter
Equation 2: Calculation of Static Thrust in Ounces
Ct d n2 D4
f = —————————— where
1,000
f = thrust in ounces, a force
d = air density in grams per liter (The word processor I had for
writing Prop Talk could not use Greek letters.)
Ct = the thrust coefficient, a dimensionless coefficient
n = propeller rpm
D = propeller diameter in meters (inches/39.37)
(The factor in the denominator is actually 1002.3 and is a constant
that contains all the unit conversions. I rounded this off to 1,000 for
ease of use in calculations.)
Another form of Equation 2 that may be easier to use on a simple
four-function calculator is:
Equation 3: Calculation of Static Thrust in Ounces
f = 4.1586 Ct d n12 D14 where
f, Ct, d = the same as in Equation 2
n1 = rpm divided by 1,000
D1 = diameter in inches/10
Equation 4: Calculation of Air Density
(11.79) (P)
d = ——————————— where
T
d = air density in grams per liter
P = local air pressure in inches of Hg
T = temperature in degrees Kelvin (Measure the temperature in
degrees Fahrenheit and calculate this temperature from Equation 5.)
Equation 5: Calculation of Temperature in Degrees Kelvin—T deg
K
T deg K = (T deg F – 32) x (5/9) + 273.16
Equation 6: Calculation of Static-Thrust Coefficient
f
Ct = ———————————
4.1586 d n12 D14
—Donald W. Brooks
the conversion to meters and divided
the diameter by 10 and the rpm by
1,000. You can do these preliminary
calculations in your head.
Let’s assume you have measured
your propeller’s rpm and diameter.
Now you want the air density to put
into the equation. The method for
calculating air density is Equation 4.
Once you know the air density, rpm
at full power, and propeller diameter,
you need only the static thrust to
calculate the static-thrust coefficient. If
you have a fish scale, you can hook it
between an anchor point and the
model’s tail. Then, using the propeller
for which you want the thrust
coefficient and running the engine or
motor at full power, measure the static
thrust.
Knowing that number, air density,
rpm, and propeller diameter, you can
use Equation 6 to calculate the
propeller-thrust coefficient. Once you
have the propeller coefficient, you can
easily calculate the thrust with that
propeller on any aircraft and at any
time and location, i.e., temperature and
air pressure (elevation effects), you
want using a tachometer reading and
Equation 3.
What are you waiting for? Find your
four-function calculator, get your
tachometer out of storage, round up a
thermometer and a De-Liar fish scale,
and call your local National Weather
Service person for the local barometric
air pressure or use Figure 1 to estimate
air pressure in your location.
Calibrate some propellers by
measuring the thrust under recorded
conditions. Mark each one you calibrate
with the measured thrust coefficient.
Once calibrated, make a full-power
tachometer measurement of rpm and
calculate your model’s static thrust.
Yes, it takes a little preparation.
You have to calibrate each propeller
you want to use in this process or you
must get the thrust coefficient from a
cataloged source. If you do not already
have the thrust coefficient for a
propeller, get out your fish scale, start
the propulsion system, make the
measurements, and calculate the thrust
coefficient.
As long as you use that propeller
and it is undamaged, the thrust
coefficient is good. You are now armed
with the knowledge to determine and
use the static-thrust coefficients on your
own.
Whether you get the thrust
coefficients from a cataloged source or
determine your own, get into the habit
02sig2.QXD 12/20/07 9:39 AM Page 36February 2008 37
Measuring Propeller Static-Thrust Coefficient
Let’s look at an example. With a calibrated thrust-measuring device, I
determined that a Zinger 14 x 10 propeller produces 41 ounces of thrust.
This propeller was operating at 4,500 rpm, the air temperature was 74
degrees Fahrenheit, and the barometric air pressure was 24.95 inches Hg.
What is this propeller’s thrust coefficient?
First, calculate D1 and n1.
D1 = 14 inches/10 = 1.4
n1 = 4,500 rpm/1,000 = 4.5
Second, calculate the temperature in Kelvin using Equation 5.
T (deg Kelvin) = (74 deg F – 32 deg F) (5/9) + 273.16
T = 296.5 deg Kelvin
Third, calculate the air density using Equation 4.
(11.79) (24.95)
d = ————————————— grams per liter
296.5
d = 0.9921 grams per liter
Fourth, calculate the static-thrust coefficient using Equation 6.
(41)
Ct = ——————————————————
4.1586 (0.9921) (4.5)2 (1.4)4
Ct = 0.128
We have determined that this Zinger 14 x 10 propeller’s thrust coefficient
is 0.128. We use a black permanent marker to write this figure on the
backside of the propeller. Now we can use Equation 3 to calculate the thrust
for different conditions of rpm or air density. MA
—Donald W. Brooks
of calculating your system’s static
thrust when preparing for flight. This
is particularly important if you are
preparing to fly a high-value model in
an unfamiliar location. Make sure you
obtain the current barometric air
pressure or use the flying-field
elevation to get an air-pressure
estimate from Figure 1.
If you know how much thrust is
required for your model to fly your
kind of mission, you can determine
the static thrust that is produced and
ensure that the thrust is adequate
before takeoff. It’s better to have an
aborted flight than a major repair or
total destruction of a model. You will
have greater confidence in your
aircraft’s flyability.
Great flying and fly safely! MA
Donald W. Brooks
[email protected]
Sources:
Altimeter (item 203):
Sun Company, Inc.
www.suncompany.net
(800) 441-0132
Engineering Fluid Dynamics (ISBN:
0-395-18607-2), 1975:
Houghton Mifflin Company
www.hmco.com
Prop Talk (ISBN 0-9657014-0-9),
1997:
ARPI Publishing
900 Bower Dr.
Idaho Falls ID 83404
Normark Weigh-In electronic digital
15-pound scale:
10395 Yellow Circle Dr.
Minnetonka MN 55343
(952) 933-7060
The author marks the back of his propellers with their pitch values and the
thrust and power coefficients.
OWN A MACHINE SHOP
1-800-476-4849 E xt. MA
Or Visit us at www.smithy.com
GUARANTEED To pay for itself!
FREE!
Info Kit
FREE!
Info Kit
Call
Today!
“I can fix ‘most anything. I don’t know how I lived without
my Smithy. It paid for itself in no time.”
• Easy to use – No
experience
required.
• Versatile – Fix or
make almost anything.
• Affordable-- 7
models starting
at $999.
• CNC Compatible
Do It Yourself on a Smithy Lathe•Mill•Drill!
02sig2.QXD 12/20/07 9:40 AM Page 37
Edition: Model Aviation - 2008/02
Page Numbers: 33,34,35,36,37,38
by Donald W. Brooks
Determining
Static Thrust
February 2008 33
Is your model’s power plant
giving you all it’s got?
I HANDED THE electric-powered GWS
Zero to Ken Marler, a long-time modeler,
and pushed the throttle to full. The motor
wound up instantly and Ken tilted the
Zero’s nose up at an angle to get an idea
of the thrust that was being produced.
“Yep,” he said with the utmost
confidence. “It’ll fly!”
And it did.
Not all modelers have such a finely
developed feel for models’ flyability.
With Scale aircraft, which tend to be on
the heavy side in terms of wing loading,
we may want more assurance than a
nose-up check. This may be particularly
true if we are competing at a flying field
that is at a different elevation from our
home field. Or the Scale model may be
too large to handle that way.
Once we know how much thrust is
required to fly a Scale model, we may
want to make sure that much thrust is
available before the next flight, which
may be conducted under different
atmospheric conditions (temperature and
air pressure). With 1/4-scale 3-D models,
we may want to assure ourselves that the
available static thrust is sufficient for
hovering or knife-edge maneuvers.
We need a way to determine static
thrust quickly and easily. You can attach
your fish scale to a rope around the
model’s tail and run the engine or motor
at full power to measure the thrust. But
determining thrust is harder if your
model does not have landing gear.
What if you could use a tachometer
reading and a four-function calculator
rather than the fish scale to determine
static thrust? Wouldn’t that be much more
convenient?
Wouldn’t it be nice to monitor staticthrust
changes as the temperature changes
and calculate thrust when you go to a
different flying field with a different
elevation? Air density is nonconstant; it
changes with temperature and with air
pressure.
I will describe an easy way to calibrate
propellers and determine the static thrust
a propeller produces from a tachometer
reading. You calibrate the propeller with
a model that has a tail wheel and then use
the propeller on any model in your fleet,
and you will still be able to calculate the
thrust that is being produced.
This article will teach you how to use
simple equations to obtain propellerthrust
coefficients and calculate static
thrust.
Sometimes it would be handy to know
the static thrust a model’s propulsion
system produces. For instance, if you
know from experience how much static
thrust is required for your model to fly
and you want to know if it would be able
to accelerate and take off, you need to
know the thrust produced. See the sidebar
about Russ Crawford’s PT-3.
If your thing is hovering or knife-edge
flight, you might even want to know
whether or not your aircraft has enough
static thrust to perform vertical
maneuvers. Barry Cazier’s Yak-54 with
its 23 x 8 propeller performs differently
in the cool morning air than it does in the
warmer mid- to late-afternoon air of a hot
day. It would be handy to determine the
Yak-54’s static thrust before an afternoon
takeoff to ensure that its performance
will be as desired.
Let’s explore takeoff and vertical
maneuvering. An old rule of thumb
floating around is that if a model’s static
thrust is equal to or greater than one-third
its weight, it will take off from a shortcut
grass field. One distributor even
published this guideline in its catalog.
I believe the rule applied to aircraft
with flat-bottom wings. A model with a
symmetrical airfoil may require more
thrust for takeoff. This rule of thumb has
worked well for me because most of my
airplanes have flat-bottomed airfoils.
I have an 84-ounce, electric-powered
Stick 40, for which the power system
produces a static thrust of 41 ounces.
That is more than 28, which is one-third
of 84. The model takes off quickly and
flies well. It will loop from level flight
but will not do vertical maneuvers. Using
the static-thrust coefficient and a
tachometer reading to determine the
available thrust can tell you if your
airplane has sufficient thrust for takeoff.
02sig2.QXD 12/20/07 9:37 AM Page 3334 MODEL AVIATION
Photos by the author Flight photos by Ron Goodwin and Barry Cazier
Left: Russ Crawford uses a Tower
Hobbies tachometer to determine his
PT-3’s rpm measurements.
Above: After testing, thrust for the PT-3
went from 25 to 28 ounces—a 12% gain.
Three of these
four instruments
are essential for
determining the
thrust coefficient.
Figure 1 can be
used in place of the
hiking barometer if
need be. The
barometer is
optional.
A 15-pound fish
scale is adequate
for models up to
120 size or 11-
pound 3-D aircraft.
Figure 1: Variation of Air Pressure
with Flying Field Elevation
(U.S. Standard Atmosphere - 1962)
20
25
30
0 2000 4000 6000 8000 10000
Elevation
(feet above sea level)
Air Pressure
(Inches of Hg)
When powering a new model, one
approach is to ensure that a certain
number of watts per pound or
horsepower per pound is available from
the power system. I will not discuss that
here. However, once you have selected
the power system, you can use the
propeller static-thrust coefficient and a
tachometer reading to determine the
thrust that is available from your
model’s power system to ensure that the
performance is adequate.
The ability to accelerate to flight
speeds, hover a model, or fly knife-edge
maneuvers depends on propeller thrust.
An extremely powerful motor or engine
that is poorly matched to a propeller
may produce less-than-desired thrust.
The Yak-54 Barry Cazier flies can do
vertical maneuvers. When we tested
Barry’s power system with its NX 23 x
8 propeller, we determined the thrust
coefficient to be 0.054. It produced 18
pounds of thrust to pull the 16.5-pound
model vertically straight up. Once we
had measured the static-thrust
02sig2.QXD 12/20/07 9:37 AM Page 34February 2008 35
Why Concern Yourself With Static Thrust?
1) Some Scale models, such as Russ Crawford’s PT-3 shown in one of
the photographs, are flying close to the limit for available thrust and
wing loading. The model may perform well at a 4,700-foot elevation in
the cool morning air but not be able to take off in the afternoon when
the air is 30° warmer.
Being able to determine the thrust with a tachometer reading helps.
Russ knows that if the PT-3’s static thrust drops below 25 ounces, he
should ground the model and come back to fly another day.
2) A Scale model may fly well at sea level outside Pensacola, Florida.
Take that same airplane to Reno, Nevada, to fly in a competition, and it
may not fly or may not fly as well because of the lower air density
caused by the reduction in air pressure. You may need to change the
propeller to get improved thrust performance at the higher elevation.
If you know the thrust coefficient and have a tachometer, you can
determine the available thrust and make necessary adjustments. Going
from sea level to 4,500 feet along with a 30° increase in temperature
may lower the air density by 20%. The decreased density affects wing
lift, propeller thrust, and engine performance.
The cumulative effects may give your model only 50% of the
performance it had at sea level. If this is your situation, you need to know
how much thrust you can depend on before the first flight at altitude.
3) Barry Cazier’s Yak-54 performs differently in the morning than in
the afternoon because of the temperature effect on air density. He likes
to fly vertical maneuvers such as the Hover, the Waterfall, and knifeedge
flight.
When we measured the thrust produced, it was approximately 18
pounds for a 16.5-pound aircraft. Barry had roughly a 9% excess of
thrust. Once he has the thrust coefficient, he can tach the propeller at
full throttle and determine quickly how much thrust he has to maneuver
the model.
4) A CL flier looks for consistent lap times for performing aerobatics
with a model. Changing the flying-field elevation or air temperatures
may necessitate a propeller change.
Lee Powell, a modeler in the local flying club, lives in Idaho Falls,
Idaho (elevation 4,740 feet), and competes in Oregon (elevation roughly
400 feet). He flies with a shallower-pitched propeller on his CL model at
the lower elevation to adjust the engine rpm, propeller thrust, and pitch
speed for consistent lap times. MA
—Donald W. Brooks
coefficient, Barry had the option of
using that propeller-thrust coefficient
and a tachometer reading to determine
the available thrust.
Basic Static-Thrust Equation: I use a
basic equation for propeller static thrust
that was provided on page 447 of an
engineering text by John A. Roberson
and Clayton T. Crowe, titled
Engineering Fluid Dynamics. The
authors were professors at Washington
State University, Pullman.
In that textbook they give the staticthrust
equation for aircraft propellers. I
experimented with model propellers
using that equation and verified that it
works just as well for 1:6-scale aircraft
propellers as it does for the 1:1
versions. And the equation works as
well now as it did in 1975; its basic
physics have not changed within the
scale range or with time.
The equation gives us a tool to
determine a propeller’s static thrust if
we know its thrust coefficient, the air
density, the operating propeller’s rpm,
and the propeller’s diameter. It is
shown in a sidebar as Equation 1.
The only hard things to obtain in
that equation are the propeller
coefficient and the air density. The
propeller-thrust coefficient must be
measured or obtained from a cataloged
source, such as my book Prop Talk.
You can calculate the air density if
you know the air pressure and
temperature. There are three ways to
get the air pressure:
• Call the local weather station and ask
for the local barometric air pressure.
• Read the value on a hiking barometer
such as the one Sun Corporation
markets.
• Use Figure 1 to estimate the air
pressure.
Figure 1 shows air pressure vs.
elevation for the U.S. Standard
Atmosphere of 1962. That figure will
also give you a working estimate for
locations where you might know the
elevation but do not have a barometer
or weather person to contact.
You can get the temperature from
your thermometer. You get the rpm
using a tachometer with your engine or
motor running at full power. Measure
the propeller’s diameter or read it off
the propeller.
I adapted Equation 1 to get it in
units that made it easy to use. The
result is Equation 2. Another form of
Equation 2 that may be easier to use on
a simple four-function calculator is
Equation 3. For it I simply eliminated
Barry Cazier’s Yak-54 flies a knife-edge pass. This maneuver
requires a significant amount of propeller thrust.
02sig2.QXD 12/20/07 9:39 AM Page 3536 MODEL AVIATION
Static-Thrust Equations
Equation 1: General Equation for Static Thrust
f = Ct r n2 D4 where
Ct = thrust coefficient for the propeller
r = density of air in which the propeller is working
n = propeller rpm
D = propeller diameter
Equation 2: Calculation of Static Thrust in Ounces
Ct d n2 D4
f = —————————— where
1,000
f = thrust in ounces, a force
d = air density in grams per liter (The word processor I had for
writing Prop Talk could not use Greek letters.)
Ct = the thrust coefficient, a dimensionless coefficient
n = propeller rpm
D = propeller diameter in meters (inches/39.37)
(The factor in the denominator is actually 1002.3 and is a constant
that contains all the unit conversions. I rounded this off to 1,000 for
ease of use in calculations.)
Another form of Equation 2 that may be easier to use on a simple
four-function calculator is:
Equation 3: Calculation of Static Thrust in Ounces
f = 4.1586 Ct d n12 D14 where
f, Ct, d = the same as in Equation 2
n1 = rpm divided by 1,000
D1 = diameter in inches/10
Equation 4: Calculation of Air Density
(11.79) (P)
d = ——————————— where
T
d = air density in grams per liter
P = local air pressure in inches of Hg
T = temperature in degrees Kelvin (Measure the temperature in
degrees Fahrenheit and calculate this temperature from Equation 5.)
Equation 5: Calculation of Temperature in Degrees Kelvin—T deg
K
T deg K = (T deg F – 32) x (5/9) + 273.16
Equation 6: Calculation of Static-Thrust Coefficient
f
Ct = ———————————
4.1586 d n12 D14
—Donald W. Brooks
the conversion to meters and divided
the diameter by 10 and the rpm by
1,000. You can do these preliminary
calculations in your head.
Let’s assume you have measured
your propeller’s rpm and diameter.
Now you want the air density to put
into the equation. The method for
calculating air density is Equation 4.
Once you know the air density, rpm
at full power, and propeller diameter,
you need only the static thrust to
calculate the static-thrust coefficient. If
you have a fish scale, you can hook it
between an anchor point and the
model’s tail. Then, using the propeller
for which you want the thrust
coefficient and running the engine or
motor at full power, measure the static
thrust.
Knowing that number, air density,
rpm, and propeller diameter, you can
use Equation 6 to calculate the
propeller-thrust coefficient. Once you
have the propeller coefficient, you can
easily calculate the thrust with that
propeller on any aircraft and at any
time and location, i.e., temperature and
air pressure (elevation effects), you
want using a tachometer reading and
Equation 3.
What are you waiting for? Find your
four-function calculator, get your
tachometer out of storage, round up a
thermometer and a De-Liar fish scale,
and call your local National Weather
Service person for the local barometric
air pressure or use Figure 1 to estimate
air pressure in your location.
Calibrate some propellers by
measuring the thrust under recorded
conditions. Mark each one you calibrate
with the measured thrust coefficient.
Once calibrated, make a full-power
tachometer measurement of rpm and
calculate your model’s static thrust.
Yes, it takes a little preparation.
You have to calibrate each propeller
you want to use in this process or you
must get the thrust coefficient from a
cataloged source. If you do not already
have the thrust coefficient for a
propeller, get out your fish scale, start
the propulsion system, make the
measurements, and calculate the thrust
coefficient.
As long as you use that propeller
and it is undamaged, the thrust
coefficient is good. You are now armed
with the knowledge to determine and
use the static-thrust coefficients on your
own.
Whether you get the thrust
coefficients from a cataloged source or
determine your own, get into the habit
02sig2.QXD 12/20/07 9:39 AM Page 36February 2008 37
Measuring Propeller Static-Thrust Coefficient
Let’s look at an example. With a calibrated thrust-measuring device, I
determined that a Zinger 14 x 10 propeller produces 41 ounces of thrust.
This propeller was operating at 4,500 rpm, the air temperature was 74
degrees Fahrenheit, and the barometric air pressure was 24.95 inches Hg.
What is this propeller’s thrust coefficient?
First, calculate D1 and n1.
D1 = 14 inches/10 = 1.4
n1 = 4,500 rpm/1,000 = 4.5
Second, calculate the temperature in Kelvin using Equation 5.
T (deg Kelvin) = (74 deg F – 32 deg F) (5/9) + 273.16
T = 296.5 deg Kelvin
Third, calculate the air density using Equation 4.
(11.79) (24.95)
d = ————————————— grams per liter
296.5
d = 0.9921 grams per liter
Fourth, calculate the static-thrust coefficient using Equation 6.
(41)
Ct = ——————————————————
4.1586 (0.9921) (4.5)2 (1.4)4
Ct = 0.128
We have determined that this Zinger 14 x 10 propeller’s thrust coefficient
is 0.128. We use a black permanent marker to write this figure on the
backside of the propeller. Now we can use Equation 3 to calculate the thrust
for different conditions of rpm or air density. MA
—Donald W. Brooks
of calculating your system’s static
thrust when preparing for flight. This
is particularly important if you are
preparing to fly a high-value model in
an unfamiliar location. Make sure you
obtain the current barometric air
pressure or use the flying-field
elevation to get an air-pressure
estimate from Figure 1.
If you know how much thrust is
required for your model to fly your
kind of mission, you can determine
the static thrust that is produced and
ensure that the thrust is adequate
before takeoff. It’s better to have an
aborted flight than a major repair or
total destruction of a model. You will
have greater confidence in your
aircraft’s flyability.
Great flying and fly safely! MA
Donald W. Brooks
[email protected]
Sources:
Altimeter (item 203):
Sun Company, Inc.
www.suncompany.net
(800) 441-0132
Engineering Fluid Dynamics (ISBN:
0-395-18607-2), 1975:
Houghton Mifflin Company
www.hmco.com
Prop Talk (ISBN 0-9657014-0-9),
1997:
ARPI Publishing
900 Bower Dr.
Idaho Falls ID 83404
Normark Weigh-In electronic digital
15-pound scale:
10395 Yellow Circle Dr.
Minnetonka MN 55343
(952) 933-7060
The author marks the back of his propellers with their pitch values and the
thrust and power coefficients.
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1-800-476-4849 E xt. MA
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02sig2.QXD 12/20/07 9:40 AM Page 37
Edition: Model Aviation - 2008/02
Page Numbers: 33,34,35,36,37,38
by Donald W. Brooks
Determining
Static Thrust
February 2008 33
Is your model’s power plant
giving you all it’s got?
I HANDED THE electric-powered GWS
Zero to Ken Marler, a long-time modeler,
and pushed the throttle to full. The motor
wound up instantly and Ken tilted the
Zero’s nose up at an angle to get an idea
of the thrust that was being produced.
“Yep,” he said with the utmost
confidence. “It’ll fly!”
And it did.
Not all modelers have such a finely
developed feel for models’ flyability.
With Scale aircraft, which tend to be on
the heavy side in terms of wing loading,
we may want more assurance than a
nose-up check. This may be particularly
true if we are competing at a flying field
that is at a different elevation from our
home field. Or the Scale model may be
too large to handle that way.
Once we know how much thrust is
required to fly a Scale model, we may
want to make sure that much thrust is
available before the next flight, which
may be conducted under different
atmospheric conditions (temperature and
air pressure). With 1/4-scale 3-D models,
we may want to assure ourselves that the
available static thrust is sufficient for
hovering or knife-edge maneuvers.
We need a way to determine static
thrust quickly and easily. You can attach
your fish scale to a rope around the
model’s tail and run the engine or motor
at full power to measure the thrust. But
determining thrust is harder if your
model does not have landing gear.
What if you could use a tachometer
reading and a four-function calculator
rather than the fish scale to determine
static thrust? Wouldn’t that be much more
convenient?
Wouldn’t it be nice to monitor staticthrust
changes as the temperature changes
and calculate thrust when you go to a
different flying field with a different
elevation? Air density is nonconstant; it
changes with temperature and with air
pressure.
I will describe an easy way to calibrate
propellers and determine the static thrust
a propeller produces from a tachometer
reading. You calibrate the propeller with
a model that has a tail wheel and then use
the propeller on any model in your fleet,
and you will still be able to calculate the
thrust that is being produced.
This article will teach you how to use
simple equations to obtain propellerthrust
coefficients and calculate static
thrust.
Sometimes it would be handy to know
the static thrust a model’s propulsion
system produces. For instance, if you
know from experience how much static
thrust is required for your model to fly
and you want to know if it would be able
to accelerate and take off, you need to
know the thrust produced. See the sidebar
about Russ Crawford’s PT-3.
If your thing is hovering or knife-edge
flight, you might even want to know
whether or not your aircraft has enough
static thrust to perform vertical
maneuvers. Barry Cazier’s Yak-54 with
its 23 x 8 propeller performs differently
in the cool morning air than it does in the
warmer mid- to late-afternoon air of a hot
day. It would be handy to determine the
Yak-54’s static thrust before an afternoon
takeoff to ensure that its performance
will be as desired.
Let’s explore takeoff and vertical
maneuvering. An old rule of thumb
floating around is that if a model’s static
thrust is equal to or greater than one-third
its weight, it will take off from a shortcut
grass field. One distributor even
published this guideline in its catalog.
I believe the rule applied to aircraft
with flat-bottom wings. A model with a
symmetrical airfoil may require more
thrust for takeoff. This rule of thumb has
worked well for me because most of my
airplanes have flat-bottomed airfoils.
I have an 84-ounce, electric-powered
Stick 40, for which the power system
produces a static thrust of 41 ounces.
That is more than 28, which is one-third
of 84. The model takes off quickly and
flies well. It will loop from level flight
but will not do vertical maneuvers. Using
the static-thrust coefficient and a
tachometer reading to determine the
available thrust can tell you if your
airplane has sufficient thrust for takeoff.
02sig2.QXD 12/20/07 9:37 AM Page 3334 MODEL AVIATION
Photos by the author Flight photos by Ron Goodwin and Barry Cazier
Left: Russ Crawford uses a Tower
Hobbies tachometer to determine his
PT-3’s rpm measurements.
Above: After testing, thrust for the PT-3
went from 25 to 28 ounces—a 12% gain.
Three of these
four instruments
are essential for
determining the
thrust coefficient.
Figure 1 can be
used in place of the
hiking barometer if
need be. The
barometer is
optional.
A 15-pound fish
scale is adequate
for models up to
120 size or 11-
pound 3-D aircraft.
Figure 1: Variation of Air Pressure
with Flying Field Elevation
(U.S. Standard Atmosphere - 1962)
20
25
30
0 2000 4000 6000 8000 10000
Elevation
(feet above sea level)
Air Pressure
(Inches of Hg)
When powering a new model, one
approach is to ensure that a certain
number of watts per pound or
horsepower per pound is available from
the power system. I will not discuss that
here. However, once you have selected
the power system, you can use the
propeller static-thrust coefficient and a
tachometer reading to determine the
thrust that is available from your
model’s power system to ensure that the
performance is adequate.
The ability to accelerate to flight
speeds, hover a model, or fly knife-edge
maneuvers depends on propeller thrust.
An extremely powerful motor or engine
that is poorly matched to a propeller
may produce less-than-desired thrust.
The Yak-54 Barry Cazier flies can do
vertical maneuvers. When we tested
Barry’s power system with its NX 23 x
8 propeller, we determined the thrust
coefficient to be 0.054. It produced 18
pounds of thrust to pull the 16.5-pound
model vertically straight up. Once we
had measured the static-thrust
02sig2.QXD 12/20/07 9:37 AM Page 34February 2008 35
Why Concern Yourself With Static Thrust?
1) Some Scale models, such as Russ Crawford’s PT-3 shown in one of
the photographs, are flying close to the limit for available thrust and
wing loading. The model may perform well at a 4,700-foot elevation in
the cool morning air but not be able to take off in the afternoon when
the air is 30° warmer.
Being able to determine the thrust with a tachometer reading helps.
Russ knows that if the PT-3’s static thrust drops below 25 ounces, he
should ground the model and come back to fly another day.
2) A Scale model may fly well at sea level outside Pensacola, Florida.
Take that same airplane to Reno, Nevada, to fly in a competition, and it
may not fly or may not fly as well because of the lower air density
caused by the reduction in air pressure. You may need to change the
propeller to get improved thrust performance at the higher elevation.
If you know the thrust coefficient and have a tachometer, you can
determine the available thrust and make necessary adjustments. Going
from sea level to 4,500 feet along with a 30° increase in temperature
may lower the air density by 20%. The decreased density affects wing
lift, propeller thrust, and engine performance.
The cumulative effects may give your model only 50% of the
performance it had at sea level. If this is your situation, you need to know
how much thrust you can depend on before the first flight at altitude.
3) Barry Cazier’s Yak-54 performs differently in the morning than in
the afternoon because of the temperature effect on air density. He likes
to fly vertical maneuvers such as the Hover, the Waterfall, and knifeedge
flight.
When we measured the thrust produced, it was approximately 18
pounds for a 16.5-pound aircraft. Barry had roughly a 9% excess of
thrust. Once he has the thrust coefficient, he can tach the propeller at
full throttle and determine quickly how much thrust he has to maneuver
the model.
4) A CL flier looks for consistent lap times for performing aerobatics
with a model. Changing the flying-field elevation or air temperatures
may necessitate a propeller change.
Lee Powell, a modeler in the local flying club, lives in Idaho Falls,
Idaho (elevation 4,740 feet), and competes in Oregon (elevation roughly
400 feet). He flies with a shallower-pitched propeller on his CL model at
the lower elevation to adjust the engine rpm, propeller thrust, and pitch
speed for consistent lap times. MA
—Donald W. Brooks
coefficient, Barry had the option of
using that propeller-thrust coefficient
and a tachometer reading to determine
the available thrust.
Basic Static-Thrust Equation: I use a
basic equation for propeller static thrust
that was provided on page 447 of an
engineering text by John A. Roberson
and Clayton T. Crowe, titled
Engineering Fluid Dynamics. The
authors were professors at Washington
State University, Pullman.
In that textbook they give the staticthrust
equation for aircraft propellers. I
experimented with model propellers
using that equation and verified that it
works just as well for 1:6-scale aircraft
propellers as it does for the 1:1
versions. And the equation works as
well now as it did in 1975; its basic
physics have not changed within the
scale range or with time.
The equation gives us a tool to
determine a propeller’s static thrust if
we know its thrust coefficient, the air
density, the operating propeller’s rpm,
and the propeller’s diameter. It is
shown in a sidebar as Equation 1.
The only hard things to obtain in
that equation are the propeller
coefficient and the air density. The
propeller-thrust coefficient must be
measured or obtained from a cataloged
source, such as my book Prop Talk.
You can calculate the air density if
you know the air pressure and
temperature. There are three ways to
get the air pressure:
• Call the local weather station and ask
for the local barometric air pressure.
• Read the value on a hiking barometer
such as the one Sun Corporation
markets.
• Use Figure 1 to estimate the air
pressure.
Figure 1 shows air pressure vs.
elevation for the U.S. Standard
Atmosphere of 1962. That figure will
also give you a working estimate for
locations where you might know the
elevation but do not have a barometer
or weather person to contact.
You can get the temperature from
your thermometer. You get the rpm
using a tachometer with your engine or
motor running at full power. Measure
the propeller’s diameter or read it off
the propeller.
I adapted Equation 1 to get it in
units that made it easy to use. The
result is Equation 2. Another form of
Equation 2 that may be easier to use on
a simple four-function calculator is
Equation 3. For it I simply eliminated
Barry Cazier’s Yak-54 flies a knife-edge pass. This maneuver
requires a significant amount of propeller thrust.
02sig2.QXD 12/20/07 9:39 AM Page 3536 MODEL AVIATION
Static-Thrust Equations
Equation 1: General Equation for Static Thrust
f = Ct r n2 D4 where
Ct = thrust coefficient for the propeller
r = density of air in which the propeller is working
n = propeller rpm
D = propeller diameter
Equation 2: Calculation of Static Thrust in Ounces
Ct d n2 D4
f = —————————— where
1,000
f = thrust in ounces, a force
d = air density in grams per liter (The word processor I had for
writing Prop Talk could not use Greek letters.)
Ct = the thrust coefficient, a dimensionless coefficient
n = propeller rpm
D = propeller diameter in meters (inches/39.37)
(The factor in the denominator is actually 1002.3 and is a constant
that contains all the unit conversions. I rounded this off to 1,000 for
ease of use in calculations.)
Another form of Equation 2 that may be easier to use on a simple
four-function calculator is:
Equation 3: Calculation of Static Thrust in Ounces
f = 4.1586 Ct d n12 D14 where
f, Ct, d = the same as in Equation 2
n1 = rpm divided by 1,000
D1 = diameter in inches/10
Equation 4: Calculation of Air Density
(11.79) (P)
d = ——————————— where
T
d = air density in grams per liter
P = local air pressure in inches of Hg
T = temperature in degrees Kelvin (Measure the temperature in
degrees Fahrenheit and calculate this temperature from Equation 5.)
Equation 5: Calculation of Temperature in Degrees Kelvin—T deg
K
T deg K = (T deg F – 32) x (5/9) + 273.16
Equation 6: Calculation of Static-Thrust Coefficient
f
Ct = ———————————
4.1586 d n12 D14
—Donald W. Brooks
the conversion to meters and divided
the diameter by 10 and the rpm by
1,000. You can do these preliminary
calculations in your head.
Let’s assume you have measured
your propeller’s rpm and diameter.
Now you want the air density to put
into the equation. The method for
calculating air density is Equation 4.
Once you know the air density, rpm
at full power, and propeller diameter,
you need only the static thrust to
calculate the static-thrust coefficient. If
you have a fish scale, you can hook it
between an anchor point and the
model’s tail. Then, using the propeller
for which you want the thrust
coefficient and running the engine or
motor at full power, measure the static
thrust.
Knowing that number, air density,
rpm, and propeller diameter, you can
use Equation 6 to calculate the
propeller-thrust coefficient. Once you
have the propeller coefficient, you can
easily calculate the thrust with that
propeller on any aircraft and at any
time and location, i.e., temperature and
air pressure (elevation effects), you
want using a tachometer reading and
Equation 3.
What are you waiting for? Find your
four-function calculator, get your
tachometer out of storage, round up a
thermometer and a De-Liar fish scale,
and call your local National Weather
Service person for the local barometric
air pressure or use Figure 1 to estimate
air pressure in your location.
Calibrate some propellers by
measuring the thrust under recorded
conditions. Mark each one you calibrate
with the measured thrust coefficient.
Once calibrated, make a full-power
tachometer measurement of rpm and
calculate your model’s static thrust.
Yes, it takes a little preparation.
You have to calibrate each propeller
you want to use in this process or you
must get the thrust coefficient from a
cataloged source. If you do not already
have the thrust coefficient for a
propeller, get out your fish scale, start
the propulsion system, make the
measurements, and calculate the thrust
coefficient.
As long as you use that propeller
and it is undamaged, the thrust
coefficient is good. You are now armed
with the knowledge to determine and
use the static-thrust coefficients on your
own.
Whether you get the thrust
coefficients from a cataloged source or
determine your own, get into the habit
02sig2.QXD 12/20/07 9:39 AM Page 36February 2008 37
Measuring Propeller Static-Thrust Coefficient
Let’s look at an example. With a calibrated thrust-measuring device, I
determined that a Zinger 14 x 10 propeller produces 41 ounces of thrust.
This propeller was operating at 4,500 rpm, the air temperature was 74
degrees Fahrenheit, and the barometric air pressure was 24.95 inches Hg.
What is this propeller’s thrust coefficient?
First, calculate D1 and n1.
D1 = 14 inches/10 = 1.4
n1 = 4,500 rpm/1,000 = 4.5
Second, calculate the temperature in Kelvin using Equation 5.
T (deg Kelvin) = (74 deg F – 32 deg F) (5/9) + 273.16
T = 296.5 deg Kelvin
Third, calculate the air density using Equation 4.
(11.79) (24.95)
d = ————————————— grams per liter
296.5
d = 0.9921 grams per liter
Fourth, calculate the static-thrust coefficient using Equation 6.
(41)
Ct = ——————————————————
4.1586 (0.9921) (4.5)2 (1.4)4
Ct = 0.128
We have determined that this Zinger 14 x 10 propeller’s thrust coefficient
is 0.128. We use a black permanent marker to write this figure on the
backside of the propeller. Now we can use Equation 3 to calculate the thrust
for different conditions of rpm or air density. MA
—Donald W. Brooks
of calculating your system’s static
thrust when preparing for flight. This
is particularly important if you are
preparing to fly a high-value model in
an unfamiliar location. Make sure you
obtain the current barometric air
pressure or use the flying-field
elevation to get an air-pressure
estimate from Figure 1.
If you know how much thrust is
required for your model to fly your
kind of mission, you can determine
the static thrust that is produced and
ensure that the thrust is adequate
before takeoff. It’s better to have an
aborted flight than a major repair or
total destruction of a model. You will
have greater confidence in your
aircraft’s flyability.
Great flying and fly safely! MA
Donald W. Brooks
[email protected]
Sources:
Altimeter (item 203):
Sun Company, Inc.
www.suncompany.net
(800) 441-0132
Engineering Fluid Dynamics (ISBN:
0-395-18607-2), 1975:
Houghton Mifflin Company
www.hmco.com
Prop Talk (ISBN 0-9657014-0-9),
1997:
ARPI Publishing
900 Bower Dr.
Idaho Falls ID 83404
Normark Weigh-In electronic digital
15-pound scale:
10395 Yellow Circle Dr.
Minnetonka MN 55343
(952) 933-7060
The author marks the back of his propellers with their pitch values and the
thrust and power coefficients.
OWN A MACHINE SHOP
1-800-476-4849 E xt. MA
Or Visit us at www.smithy.com
GUARANTEED To pay for itself!
FREE!
Info Kit
FREE!
Info Kit
Call
Today!
“I can fix ‘most anything. I don’t know how I lived without
my Smithy. It paid for itself in no time.”
• Easy to use – No
experience
required.
• Versatile – Fix or
make almost anything.
• Affordable-- 7
models starting
at $999.
• CNC Compatible
Do It Yourself on a Smithy Lathe•Mill•Drill!
02sig2.QXD 12/20/07 9:40 AM Page 37
Edition: Model Aviation - 2008/02
Page Numbers: 33,34,35,36,37,38
by Donald W. Brooks
Determining
Static Thrust
February 2008 33
Is your model’s power plant
giving you all it’s got?
I HANDED THE electric-powered GWS
Zero to Ken Marler, a long-time modeler,
and pushed the throttle to full. The motor
wound up instantly and Ken tilted the
Zero’s nose up at an angle to get an idea
of the thrust that was being produced.
“Yep,” he said with the utmost
confidence. “It’ll fly!”
And it did.
Not all modelers have such a finely
developed feel for models’ flyability.
With Scale aircraft, which tend to be on
the heavy side in terms of wing loading,
we may want more assurance than a
nose-up check. This may be particularly
true if we are competing at a flying field
that is at a different elevation from our
home field. Or the Scale model may be
too large to handle that way.
Once we know how much thrust is
required to fly a Scale model, we may
want to make sure that much thrust is
available before the next flight, which
may be conducted under different
atmospheric conditions (temperature and
air pressure). With 1/4-scale 3-D models,
we may want to assure ourselves that the
available static thrust is sufficient for
hovering or knife-edge maneuvers.
We need a way to determine static
thrust quickly and easily. You can attach
your fish scale to a rope around the
model’s tail and run the engine or motor
at full power to measure the thrust. But
determining thrust is harder if your
model does not have landing gear.
What if you could use a tachometer
reading and a four-function calculator
rather than the fish scale to determine
static thrust? Wouldn’t that be much more
convenient?
Wouldn’t it be nice to monitor staticthrust
changes as the temperature changes
and calculate thrust when you go to a
different flying field with a different
elevation? Air density is nonconstant; it
changes with temperature and with air
pressure.
I will describe an easy way to calibrate
propellers and determine the static thrust
a propeller produces from a tachometer
reading. You calibrate the propeller with
a model that has a tail wheel and then use
the propeller on any model in your fleet,
and you will still be able to calculate the
thrust that is being produced.
This article will teach you how to use
simple equations to obtain propellerthrust
coefficients and calculate static
thrust.
Sometimes it would be handy to know
the static thrust a model’s propulsion
system produces. For instance, if you
know from experience how much static
thrust is required for your model to fly
and you want to know if it would be able
to accelerate and take off, you need to
know the thrust produced. See the sidebar
about Russ Crawford’s PT-3.
If your thing is hovering or knife-edge
flight, you might even want to know
whether or not your aircraft has enough
static thrust to perform vertical
maneuvers. Barry Cazier’s Yak-54 with
its 23 x 8 propeller performs differently
in the cool morning air than it does in the
warmer mid- to late-afternoon air of a hot
day. It would be handy to determine the
Yak-54’s static thrust before an afternoon
takeoff to ensure that its performance
will be as desired.
Let’s explore takeoff and vertical
maneuvering. An old rule of thumb
floating around is that if a model’s static
thrust is equal to or greater than one-third
its weight, it will take off from a shortcut
grass field. One distributor even
published this guideline in its catalog.
I believe the rule applied to aircraft
with flat-bottom wings. A model with a
symmetrical airfoil may require more
thrust for takeoff. This rule of thumb has
worked well for me because most of my
airplanes have flat-bottomed airfoils.
I have an 84-ounce, electric-powered
Stick 40, for which the power system
produces a static thrust of 41 ounces.
That is more than 28, which is one-third
of 84. The model takes off quickly and
flies well. It will loop from level flight
but will not do vertical maneuvers. Using
the static-thrust coefficient and a
tachometer reading to determine the
available thrust can tell you if your
airplane has sufficient thrust for takeoff.
02sig2.QXD 12/20/07 9:37 AM Page 3334 MODEL AVIATION
Photos by the author Flight photos by Ron Goodwin and Barry Cazier
Left: Russ Crawford uses a Tower
Hobbies tachometer to determine his
PT-3’s rpm measurements.
Above: After testing, thrust for the PT-3
went from 25 to 28 ounces—a 12% gain.
Three of these
four instruments
are essential for
determining the
thrust coefficient.
Figure 1 can be
used in place of the
hiking barometer if
need be. The
barometer is
optional.
A 15-pound fish
scale is adequate
for models up to
120 size or 11-
pound 3-D aircraft.
Figure 1: Variation of Air Pressure
with Flying Field Elevation
(U.S. Standard Atmosphere - 1962)
20
25
30
0 2000 4000 6000 8000 10000
Elevation
(feet above sea level)
Air Pressure
(Inches of Hg)
When powering a new model, one
approach is to ensure that a certain
number of watts per pound or
horsepower per pound is available from
the power system. I will not discuss that
here. However, once you have selected
the power system, you can use the
propeller static-thrust coefficient and a
tachometer reading to determine the
thrust that is available from your
model’s power system to ensure that the
performance is adequate.
The ability to accelerate to flight
speeds, hover a model, or fly knife-edge
maneuvers depends on propeller thrust.
An extremely powerful motor or engine
that is poorly matched to a propeller
may produce less-than-desired thrust.
The Yak-54 Barry Cazier flies can do
vertical maneuvers. When we tested
Barry’s power system with its NX 23 x
8 propeller, we determined the thrust
coefficient to be 0.054. It produced 18
pounds of thrust to pull the 16.5-pound
model vertically straight up. Once we
had measured the static-thrust
02sig2.QXD 12/20/07 9:37 AM Page 34February 2008 35
Why Concern Yourself With Static Thrust?
1) Some Scale models, such as Russ Crawford’s PT-3 shown in one of
the photographs, are flying close to the limit for available thrust and
wing loading. The model may perform well at a 4,700-foot elevation in
the cool morning air but not be able to take off in the afternoon when
the air is 30° warmer.
Being able to determine the thrust with a tachometer reading helps.
Russ knows that if the PT-3’s static thrust drops below 25 ounces, he
should ground the model and come back to fly another day.
2) A Scale model may fly well at sea level outside Pensacola, Florida.
Take that same airplane to Reno, Nevada, to fly in a competition, and it
may not fly or may not fly as well because of the lower air density
caused by the reduction in air pressure. You may need to change the
propeller to get improved thrust performance at the higher elevation.
If you know the thrust coefficient and have a tachometer, you can
determine the available thrust and make necessary adjustments. Going
from sea level to 4,500 feet along with a 30° increase in temperature
may lower the air density by 20%. The decreased density affects wing
lift, propeller thrust, and engine performance.
The cumulative effects may give your model only 50% of the
performance it had at sea level. If this is your situation, you need to know
how much thrust you can depend on before the first flight at altitude.
3) Barry Cazier’s Yak-54 performs differently in the morning than in
the afternoon because of the temperature effect on air density. He likes
to fly vertical maneuvers such as the Hover, the Waterfall, and knifeedge
flight.
When we measured the thrust produced, it was approximately 18
pounds for a 16.5-pound aircraft. Barry had roughly a 9% excess of
thrust. Once he has the thrust coefficient, he can tach the propeller at
full throttle and determine quickly how much thrust he has to maneuver
the model.
4) A CL flier looks for consistent lap times for performing aerobatics
with a model. Changing the flying-field elevation or air temperatures
may necessitate a propeller change.
Lee Powell, a modeler in the local flying club, lives in Idaho Falls,
Idaho (elevation 4,740 feet), and competes in Oregon (elevation roughly
400 feet). He flies with a shallower-pitched propeller on his CL model at
the lower elevation to adjust the engine rpm, propeller thrust, and pitch
speed for consistent lap times. MA
—Donald W. Brooks
coefficient, Barry had the option of
using that propeller-thrust coefficient
and a tachometer reading to determine
the available thrust.
Basic Static-Thrust Equation: I use a
basic equation for propeller static thrust
that was provided on page 447 of an
engineering text by John A. Roberson
and Clayton T. Crowe, titled
Engineering Fluid Dynamics. The
authors were professors at Washington
State University, Pullman.
In that textbook they give the staticthrust
equation for aircraft propellers. I
experimented with model propellers
using that equation and verified that it
works just as well for 1:6-scale aircraft
propellers as it does for the 1:1
versions. And the equation works as
well now as it did in 1975; its basic
physics have not changed within the
scale range or with time.
The equation gives us a tool to
determine a propeller’s static thrust if
we know its thrust coefficient, the air
density, the operating propeller’s rpm,
and the propeller’s diameter. It is
shown in a sidebar as Equation 1.
The only hard things to obtain in
that equation are the propeller
coefficient and the air density. The
propeller-thrust coefficient must be
measured or obtained from a cataloged
source, such as my book Prop Talk.
You can calculate the air density if
you know the air pressure and
temperature. There are three ways to
get the air pressure:
• Call the local weather station and ask
for the local barometric air pressure.
• Read the value on a hiking barometer
such as the one Sun Corporation
markets.
• Use Figure 1 to estimate the air
pressure.
Figure 1 shows air pressure vs.
elevation for the U.S. Standard
Atmosphere of 1962. That figure will
also give you a working estimate for
locations where you might know the
elevation but do not have a barometer
or weather person to contact.
You can get the temperature from
your thermometer. You get the rpm
using a tachometer with your engine or
motor running at full power. Measure
the propeller’s diameter or read it off
the propeller.
I adapted Equation 1 to get it in
units that made it easy to use. The
result is Equation 2. Another form of
Equation 2 that may be easier to use on
a simple four-function calculator is
Equation 3. For it I simply eliminated
Barry Cazier’s Yak-54 flies a knife-edge pass. This maneuver
requires a significant amount of propeller thrust.
02sig2.QXD 12/20/07 9:39 AM Page 3536 MODEL AVIATION
Static-Thrust Equations
Equation 1: General Equation for Static Thrust
f = Ct r n2 D4 where
Ct = thrust coefficient for the propeller
r = density of air in which the propeller is working
n = propeller rpm
D = propeller diameter
Equation 2: Calculation of Static Thrust in Ounces
Ct d n2 D4
f = —————————— where
1,000
f = thrust in ounces, a force
d = air density in grams per liter (The word processor I had for
writing Prop Talk could not use Greek letters.)
Ct = the thrust coefficient, a dimensionless coefficient
n = propeller rpm
D = propeller diameter in meters (inches/39.37)
(The factor in the denominator is actually 1002.3 and is a constant
that contains all the unit conversions. I rounded this off to 1,000 for
ease of use in calculations.)
Another form of Equation 2 that may be easier to use on a simple
four-function calculator is:
Equation 3: Calculation of Static Thrust in Ounces
f = 4.1586 Ct d n12 D14 where
f, Ct, d = the same as in Equation 2
n1 = rpm divided by 1,000
D1 = diameter in inches/10
Equation 4: Calculation of Air Density
(11.79) (P)
d = ——————————— where
T
d = air density in grams per liter
P = local air pressure in inches of Hg
T = temperature in degrees Kelvin (Measure the temperature in
degrees Fahrenheit and calculate this temperature from Equation 5.)
Equation 5: Calculation of Temperature in Degrees Kelvin—T deg
K
T deg K = (T deg F – 32) x (5/9) + 273.16
Equation 6: Calculation of Static-Thrust Coefficient
f
Ct = ———————————
4.1586 d n12 D14
—Donald W. Brooks
the conversion to meters and divided
the diameter by 10 and the rpm by
1,000. You can do these preliminary
calculations in your head.
Let’s assume you have measured
your propeller’s rpm and diameter.
Now you want the air density to put
into the equation. The method for
calculating air density is Equation 4.
Once you know the air density, rpm
at full power, and propeller diameter,
you need only the static thrust to
calculate the static-thrust coefficient. If
you have a fish scale, you can hook it
between an anchor point and the
model’s tail. Then, using the propeller
for which you want the thrust
coefficient and running the engine or
motor at full power, measure the static
thrust.
Knowing that number, air density,
rpm, and propeller diameter, you can
use Equation 6 to calculate the
propeller-thrust coefficient. Once you
have the propeller coefficient, you can
easily calculate the thrust with that
propeller on any aircraft and at any
time and location, i.e., temperature and
air pressure (elevation effects), you
want using a tachometer reading and
Equation 3.
What are you waiting for? Find your
four-function calculator, get your
tachometer out of storage, round up a
thermometer and a De-Liar fish scale,
and call your local National Weather
Service person for the local barometric
air pressure or use Figure 1 to estimate
air pressure in your location.
Calibrate some propellers by
measuring the thrust under recorded
conditions. Mark each one you calibrate
with the measured thrust coefficient.
Once calibrated, make a full-power
tachometer measurement of rpm and
calculate your model’s static thrust.
Yes, it takes a little preparation.
You have to calibrate each propeller
you want to use in this process or you
must get the thrust coefficient from a
cataloged source. If you do not already
have the thrust coefficient for a
propeller, get out your fish scale, start
the propulsion system, make the
measurements, and calculate the thrust
coefficient.
As long as you use that propeller
and it is undamaged, the thrust
coefficient is good. You are now armed
with the knowledge to determine and
use the static-thrust coefficients on your
own.
Whether you get the thrust
coefficients from a cataloged source or
determine your own, get into the habit
02sig2.QXD 12/20/07 9:39 AM Page 36February 2008 37
Measuring Propeller Static-Thrust Coefficient
Let’s look at an example. With a calibrated thrust-measuring device, I
determined that a Zinger 14 x 10 propeller produces 41 ounces of thrust.
This propeller was operating at 4,500 rpm, the air temperature was 74
degrees Fahrenheit, and the barometric air pressure was 24.95 inches Hg.
What is this propeller’s thrust coefficient?
First, calculate D1 and n1.
D1 = 14 inches/10 = 1.4
n1 = 4,500 rpm/1,000 = 4.5
Second, calculate the temperature in Kelvin using Equation 5.
T (deg Kelvin) = (74 deg F – 32 deg F) (5/9) + 273.16
T = 296.5 deg Kelvin
Third, calculate the air density using Equation 4.
(11.79) (24.95)
d = ————————————— grams per liter
296.5
d = 0.9921 grams per liter
Fourth, calculate the static-thrust coefficient using Equation 6.
(41)
Ct = ——————————————————
4.1586 (0.9921) (4.5)2 (1.4)4
Ct = 0.128
We have determined that this Zinger 14 x 10 propeller’s thrust coefficient
is 0.128. We use a black permanent marker to write this figure on the
backside of the propeller. Now we can use Equation 3 to calculate the thrust
for different conditions of rpm or air density. MA
—Donald W. Brooks
of calculating your system’s static
thrust when preparing for flight. This
is particularly important if you are
preparing to fly a high-value model in
an unfamiliar location. Make sure you
obtain the current barometric air
pressure or use the flying-field
elevation to get an air-pressure
estimate from Figure 1.
If you know how much thrust is
required for your model to fly your
kind of mission, you can determine
the static thrust that is produced and
ensure that the thrust is adequate
before takeoff. It’s better to have an
aborted flight than a major repair or
total destruction of a model. You will
have greater confidence in your
aircraft’s flyability.
Great flying and fly safely! MA
Donald W. Brooks
[email protected]
Sources:
Altimeter (item 203):
Sun Company, Inc.
www.suncompany.net
(800) 441-0132
Engineering Fluid Dynamics (ISBN:
0-395-18607-2), 1975:
Houghton Mifflin Company
www.hmco.com
Prop Talk (ISBN 0-9657014-0-9),
1997:
ARPI Publishing
900 Bower Dr.
Idaho Falls ID 83404
Normark Weigh-In electronic digital
15-pound scale:
10395 Yellow Circle Dr.
Minnetonka MN 55343
(952) 933-7060
The author marks the back of his propellers with their pitch values and the
thrust and power coefficients.
OWN A MACHINE SHOP
1-800-476-4849 E xt. MA
Or Visit us at www.smithy.com
GUARANTEED To pay for itself!
FREE!
Info Kit
FREE!
Info Kit
Call
Today!
“I can fix ‘most anything. I don’t know how I lived without
my Smithy. It paid for itself in no time.”
• Easy to use – No
experience
required.
• Versatile – Fix or
make almost anything.
• Affordable-- 7
models starting
at $999.
• CNC Compatible
Do It Yourself on a Smithy Lathe•Mill•Drill!
02sig2.QXD 12/20/07 9:40 AM Page 37