Propellor Thrust Calculations for P-51 Mustang Simulation

[eaglestrike]

Hello everyone
This is my first post. I am currently making a simulation of a P-51 Mustang in Blender3d. ("www.blender.org"[/URL]) as the other ones are from the Wright section of the NASA website as well.
If someone could explain these in simple terms for me (as I am only a tenth Grader) (or just tell me how to get the Static pressure, as that's the main problem) it would be appreciated.

The Lift and Drag calculations are from [PLAIN]http://wright.nasa.gov/airplane/lifteq.html" .

 

[rcgldr]

Static pressure means the ambient pressure of the air at the altitude your flying your model. Sea level would be 14.7 psi, 1940 feet above sea level would be 13.7 psi, ...

I'm not sure how you will be able to measure exit velocity as the airframe will interfere with this a bit. You could measure static thrust, using some type of scale, but I don't know the accuracy you're looking for. Prop efficiency will peak at some combination of forward speed and rpm.

 

[eaglestrike]

I think ill use standard calculations for this (pressure = 101.325 kPa). Is sweep area the area covered by the prop or is it area/time? The rest is simple but I do not understand how engine power is used in these calculations. How can merely Propeller size the airflow determine velocity without any engine power being absorbed?

 

[eaglestrike]

? Anyone help? I know that this is a high-profile forum but if someone could help it would be appreciated.

 

[rcgldr]

Swept area is the area of a circle with the radius length equal center of prop axle to prop tip.

Engine power is a bit tricky. The prop creates a torque opposine engine, based on rpm of the prop, air speed, prop pitch, prop diameter, prop airfoil, ... . Usually you have to get this data from a table or graph for a specific propeller. The required engine power is equal to the prop's opposing torque times rpm. The props output power (thrust times airspeed) will be 15% to 25% less than this.

 

[eaglestrike]

Ive got the Swept area at 5.29296786914 m2. The rpm at full throttle is 2700, 3000 with WEP. I'll work with the NASA ones first, then see the rest. If anyone has some better calculations, please post them as the Drag And Lift coefficient calculations are the same but the prop thrust ones aren't.

 

[eaglestrike]

More help needed. It requires the velocity to calculate the entrance and exit velocities. But the velocity of what? I assume that the velocity is the propellor velocity which can be either Torque or m/s (circumference x rpm?). I need to be sure what type of velocity it is. If it is Torque then my question before is solved, as the engine power is incorporated into the calculation.

 

[eaglestrike]

[sigh] ill wait

 

[rcgldr]

The entry velocity is the airspeed of the aircraft. The exit velocity is the speed of the air at the moment it's pressure returns to ambient.

What happens is air accelerates and decreases in pressure in front of the prop, then mostly experiences an increase in pressure to above ambient without much change in velocity across the "prop disk". The air then continues to accelerate again as it's pressure decreases back to ambient. The velocity of the air at the prop is about the average of the entry and exit velocities.

You need the specific data for the prop, number of blades, prop diameter, prop pitch, rpm, ... So far you have the diameter, and the rpm. For a P-51 the number of blades is 4.

The specs I get for a P51D shows 1490hp (at takeoff) 1590 hp (full power) to 1690hp (War Emergency Power?). The engine is called a 1650 for 1650 hp, but I don't know under what conditions). The propeller is a constant speed (variable pitch) propeller, 11 feet 2 inches in diameter with 4 blades.

I did a web search and the only info I could find was a "guestimate" of 3000 lbs of thrust at lower air speeds at 25,000 feet with 1200 hp.

 

[redargon]

Are you trying to calculate the thrust for different velocities or do you need a maximum thrust or are you trying to equate thrust with some other variable? I can't get onto the NASA page to check the calcs there.

 

[rcgldr]

If you guestimate that the prop is 80% efficient, then with 1590 hp input (full throttle), then you get .8 x 1590 hp = 1272 hp output.

Note, the engine is geared, 3000 rpm translates into 1437 rpm based on this article.
http://www.airnews.co.za/home/index.php?option=com_content&view=article&id=340:flying-the-p-51d-mustang&catid=83:aircrafteviews&Itemid=67

The math here is simply based on power and density of air.

Let Vp = velocity of air at the prop
Let Ve = velocity of air downstream of prop (when it's pressure returns to ambient)
Let Mf = mass flow of air
Area swept by prop is (pi x ((11 ft 2 in) / 2)^2) = 97.935 ft^2

1272 hp = thrust (lbs) x Vp (ft / sec) / 550

thrust = 1272 x 550 / (Vp) = Mf * (V_e- V₀)

Volume flow = Vp (ft/sec) x 97.935 (ft^2)
(Using Ve doesn't work because the cross section is smaller than the prop swept
area and unknown).

Ignoring compression effects for mass flow:
Mf = volume flow (ft^3 / sec) * .002330 slug / ft^3 (for air temp around 65 degrees)

Mf = Vp x 97.935 * .002330 slug / sec

thrust = 1272 x 550 / Vp = Mf * (Ve - V₀)

thrust = 1272 x 550 / Vp = Vp x 97.935 * .002330 * Ve

For static thrust, V₀ = 0

Ve = 2 Vp

thrust = 1272 x 550 / Vp = Vp x 97.935 * .002330 * 2 * Vp
1272 x 550 = Vp^3 x 97.935 * .002330 * 2
Vp^3 = 1272 x 550 / (97.935 * .002330 * 2)
Vp = 115.3 ft / sec

static thrust = 6067.5 lbs
Ve = 230.6 ft / sec

The pressure delta across the prop disk for a check

delta p = thrust / area = thrust / (97.935 ft^2) = 61.95 lbs/ft^2
delta p = .5 density (V_e^2 + V₀^2) = 61.95 lbs/ft^2
delta p = .43 psi

I'm not sure of this calculation. I don't know the efficiency in a static situation. One prop calculator shows 3200 lbs of static thrust for a 134 inch diameter, 4 bladed prop at 1435 rpm, which would indicate a much lower efficiency. Another calculator shows 3350 lbs of force.

For thrust = 3300 lbs:
delta p = thrust / area = 3300 / 97.935 = 33.696 lb/ft^2 = .5 x .002330 x Ve^2
Ve = 170 ft / sec
Vp = 85 ft / sec
Output power = 510 hp, effciency = 32.1%

Note this agrees with the math shown here for the static thrust situation:

prop_efficiency.htm

To use the units from above, for power, the unit is 1 ft lb / sec

1 hp = 550 ft lb / sec

Plugging this into the static thrust equation from the web site
FOM = 32.0884821%
Pavail = 1590 hp
Power output = FOM x Pavail = 510.2068654 hp
density = .002330 slug / ft^3
area = 97.935 ft^2

.320884821 x 1590 hp = 510.2068654 hp = 280613.776 lb ft / sec

Thrust = (280613.776 / ((1/2) x (1/(.002330x97.935))^(1/2)))^(2/3)
Thrust = (280613.776 / (((1/2) x (1/(.22818855)))^(1/2)))^(2/3)
Thrust = (280613.776 / (((1/2) x (4.382340832))^(1/2)))^(2/3)
Thrust = (280613.776 / ((2.191170416)^(1/2)))^(2/3)
Thrust = (280613.776 / (1.480260253))^(2/3)
Thrust = (189570.5674)^(2/3)
Thrust = 3300 lbs
Vp = 85.03447757 ft / sec


For a given airspeed, air density, and prop efficiency, you'd have to do similar math
output power = 1590 hp x efficiency
Ve = 2Vp - V₀
Mf = Vp (ft / sec) x density_of_air (slug / ft ^3) x 97.935 ft^2

Here is another propeller math oriented link:

http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node86.html

Another thing to note is that assuming density doesn't change, then the cross sectional area of the air affected by a propeller decreases inversely with speed. In a static thrust situation, the cross sectional area or the prop wash at the "exit" is 1/2 the cross sectional area at the prop. This creates a paradox in this simplified math, because at zero speed in front of the prop, the cross sectional area would be infinite. Look at figure 11.25 in section 11.7.2 and note what would happen if V₀ = 0.

 

[eaglestrike]

thanks i got to look at that in detail. Unfortunately my computer died two days ago (a virus wiped out the motherboard). Luckily I kept backups. Thx ill see this in detail although converting is a bit tricky as I am using metric atm, but once these are done i can just convert ft/sec to m/s.

 

[eaglestrike]

Can you convert this to metric please?

 

[eaglestrike]

Heres what I got so far (after about 10 mins):
At 80% efficiency of 1111kW of power, 0.8 * 1111 = 888.8kW
Engine is geared at a ratio of 479:1000

Vp = Velocity of air at the prop
Ve = Velocity of air downstream
Mf = Mass flow of the air
Swept area = 3.4036m2

888.8 kW = thrust(N) * Vp (m/s) /1000
## Since 1W = Force to push 1n 1m
Thrust = 888.8 * 1000 = Mf*(Ve-V0)

Mf = Vp * Swept Area * Air Density (calculated prior at 20o = 1.204 kg/m3)

Ive got a lot of end of term stuff coming, so I don't have much time. Please tell me if I am on the right track.

 

[eaglestrike]

<a while later>
I did a bit more :

Thrust = Vp^3 * 3.406 * 1.204 * 2
Vp^3 = 888.8*1000 / (3.4036 * 1.212 * 2)
Vp = 47.687m/s

the Value I am getting is higher than yours, meaning that something is wrong in my calculations.

 

[rcgldr]

Can you convert this to metric please?

Using the second lower efficiency number:

Let Vp = velocity of air at the prop
Let Ve = velocity of air downstream of prop (when it's pressure returns to ambient)
Let Mf = mass flow of air
Area swept by prop is (pi x ((11 ft 2 in) / 2)^2) = 97.935 ft^2
Let efficiency = 32.0884821%
prop diameter = 3.4036 m
prop area = 9.0984395 m^2

engine power = 1185662.8 watts
prop output power = 380461 watts

380461 watts = thrust N x Vp (m/s)

thrust = 380461 / Vp = Mf * (V_e- V₀)

Volume flow = Vp (m/s) x 9.09844 (m^2)

Ignoring compression effects for mass flow:
Mf = volume flow (ft^3 / sec) * 1.200 (kg) / m^3

Mf = Vp x 9.09844 * 1.200 kg / sec

thrust = 380461 / Vp = Mf * (V_e- V₀)

thrust = 380461 / Vp = Vp x 9.09844 * 1.200 * (V_e- V₀)

For static thrust, V₀ = 0

thrust = 380461 / Vp = Vp x 9.09844 * 1.200 * Ve

Ve = 2 Vp

thrust = 380461 / Vp = Vp x 9.09844 * 1.200 * 2 * Vp
380461 = Vp^3 x 9.09844 * 1.200 * 2

Vp^3 = 380461 / (9.09844 * 1.200 * 2)
Vp = 25.925 m / s (= 85 ft / s)
Ve = 51.85 m / s

static thrust = 14675 N (= 3299 lbs)

 

[eaglestrike]

Thx for the help. It is going to take me a while to understand these. 1 more question (it may sound stupid)- does rpm increase with throttle and does that count in the equation.

 

[rcgldr]

Thx for the help. It is going to take me a while to understand these. 1 more question - does rpm increase with throttle.

Depends on the prop design. If prop if fixed pitch the engine rpm will vary as normal, associated with throttle inputs. If the prop is a variable pitch, constant speed prop, then once the throttle is at or above the constant rpm level, then the rpms stay about the same, and only the prop pitch is modified. The power is increases, but all of it goes into an increase in torque and not an increase in rpm as well.

throttle - does that count in the equation.

Yes, the throttle input is related to the power. The example equation is assuming peak power.

 

[eaglestrike]

Ive understod the calculations.
The thrust is 14675N, but how can that power an aircraft with the weight of 34650N?
Does the figure of rpm come in? I've headed up an excel sheet and it all works out with the NASA calculations, but I need the calculation for overall thrust. I've got Lift and Drag which rely on Velocity, so Thrust will complete the equation and I can go on to programming stall.

 

[rcgldr]

The thrust is 14675N, but how can that power an aircraft with the weight of 34650N?

Because the lift to drag ratio is less than 14675 / 34650. I'm guessing that the lift to drag ratio for the aircraft is about 1 / 7 or lower (more efficient). This limits the climb angle (vertical would require more thrust than weight) to tan^-1(14675 / 34650) = 22.9 degrees if lift to drag ratio were zero, the actual maximum climb angle would less. For level flight, the thrust only has to overcome drag, which is much less than the weight of the aircraft.

 

[eaglestrike]

For level flight, the thrust only has to overcome drag, which is much less than the weight of the aircraft.

What I don't undersatnd is how the Aircraft gets enough speed in this equation. 14675.72745/34650 = 0.423541918m/s = 1.524kmph. Does the Linear Velocity add? I tried this in my model and It worked too fast. Does rpm come in? If so, then force = 660407.7N, giving it a velcocity of 19m/s = 68.61kmph.

 

[eaglestrike]

If so, then force = 660407.7N, giving it a velcocity of 19m/s = 68.61kmph.

All equations give the same answer, (obviously). What is the total thrust, needed to propel the aircraft?

 

[rcgldr]

Vp = 25.925 m / s
Ve = 51.85 m / s
static thrust = 14675 N

What I don't undersatnd is how the aircraft gets enough speed in this equation.

Note the Vp above, in a static situation (aircraft not moving with respect to the air), the air speed at the prop is 25.925 m/s, and continues to accelerate to 51.85 m/s as it's pressure returns to ambient. If the aircraft is moving, then the thrust will decrease, as the velocity of the air at the prop increases. Since the prop is a variable pitch, constant speed prop, the decrease in thrust won't be as much as the increase in air speed.

Perhaps at 150 m/s the thrust is down to 7,000 N, just a guess here, I have no idea on how much force there is. If you know the drag at top speed, then the thrust equals the drag at that speed.

 

[eaglestrike]

These are my equations so far in my script
rho =
p_a = 9.909844 $prop area Mass = 34650 Vp = (power / (p_a * rho * 2))**(1/3.0)$ Air Velocity at the prop
Ve = 2 * Vp ## Air Velocity downstream
Thrust (T) = power / Vp
D = Drag
Force:
y = (T-D)
z = Lift (L)

The problem is that the aircraft never actually gathers enough Force to take off. The equations given are for an aircraft already moving at a certain velocity. What I need is some simple equations that will give a Force value at all speeds and different throttle values.

I understand that you might be getting annoyed by now, so If you could refer me to an article that gives all the information I need or at least some of it then I can work it out.

 

[rcgldr]

The equations given are for an aircraft already moving at a certain velocity.

The prop is operating in it's own induced washed, even when the air craft is stationary. Ve = 2 x Vp in the equations above only occurs when aircraft velocity is zero.

 

[russ_watters]

What I don't undersatnd is how the Aircraft gets enough speed in this equation. 14675.72745/34650 = 0.423541918m/s = 1.524kmph. Does the Linear Velocity add? I tried this in my model and It worked too fast. Does rpm come in? If so, then force = 660407.7N, giving it a velcocity of 19m/s = 68.61kmph.

You are using the numbers incorrectly: f=ma. a is acceleration, not speed. And m is mass, not weight. So all the thrust and weight can tell you is how fast the plane accelerates, not what its top speed is. Top speed is determined by drag.

Thos numbers tell us...

a=f/m = 14675/(34650/9.8)=
...that the airplane accelerates at 4.15 m/s/s, which sounds reasonable to me.

 

[eaglestrike]

Thx for pointing that out. The aircraft takes off comfortably with an Efficiency of 0.8-0.9 (the FOM is too low).
I have found some graphs on the propeller efficiency/advance ratio with the propeller pitch setting.
http://www.mudpond.org/fs_props.pdf"
I could use the graphs, or I could use these equations:
http://www.aa.washington.edu/courses/aa441/07-Propellers.pdf"
What should I use?
An easier alternative is to graph the highest prop pitch settings together (as the prop pitch will be automatic anyhow) and use them for the efficiency.

 

[rcgldr]

The aircraft takes off comfortably with an efficiency of 0.8-0.9 (the FOM is too low).

At launch, the advance ratio (airspeed x pi / (prop tip speed perpendicular to airspeed)), is zero, and the efficiency is low as shown on the graphs. Also, even though power output is described as thrust versus airspeed, thrust versus induced airspeed at the prop handles the case of zero airspeed.

An easier alternative is to graph the highest prop pitch settings together (as the prop pitch will be automatic anyhow) and use them for the efficiency.

I don't know about this. The throttle setting and air speed affect the prop pitch, with a region of parameters where the prop rpm is nearly constant.

 

[eaglestrike]

Thanks alor Mr Reid. My calculations are done. I plotted the efficiency and set it with the advance ratio in my program and now thrust is fine.
This doesn't fit in with the name anymore, but could anyone help me with the roll, pitch and yaw physics now?
And how do the leaving the undercarriage and flaps affect drag? I need this for my equation as well.

 

[rcgldr]

This doesn't fit in with the name anymore, but could anyone help me with the roll, pitch and yaw physics now? How do the leaving the undercarriage and flaps affect drag? I need this for my equation as well.

I don't know how the math is done for this. The control surfaces generate a torque, and you have a combination of aerodynamic drag (large effect) and angular inertia (small effect) affecting the response rate. The prop on a P51 also generates a signifcant amount of roll torque, especially at low speeds during take offs. The "rolling" prop wash can also affect the vertical stabilizer, adding a yaw related response.

Landing gear results in slight drag and downwards pitch torque. Flaps have a lot of drag and increase downwards pitching torque. I would assume tail control surfaces have suffiencient authority that these aren't issues, but they shouldn't be used at high speed. The standard P51 didn't have dive brakes, but there were variations that did.

 

[Cyrus]

Thanks alor Mr Reid. My calculations are done. I plotted the efficiency and set it with the advance ratio in my program and now thrust is fine.
This doesn't fit in with the name anymore, but could anyone help me with the roll, pitch and yaw physics now?
And how do the leaving the undercarriage and flaps affect drag? I need this for my equation as well.

I'm currently making a flight dynamics model for two aircraft. In a nutshell, you're way over your head.