# Determining the acceleration of an aircraft

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1. Nov 21, 2015

### GeForce7

Hello everyone. I'm currently doing a research on aircraft performance for my school project, which require me to design an airfoil to suit a specific mission. I'm not very familiar with the formulas used for calculating the aircraft performance and I would like to seek help from the forum.

Using the NASA FoilSim, I came out with an airfoil profile with a maximum coefficient of lift of 1.888. Using the lift equation, I calculated the minimum speed which I must fulfill in order to gain enough lift to counter the weight which is approximately 39.18 m/s.

V_stall=√(W/(CL_max×0.5×ρ×S))
V_stall=√(39240/(1.888×0.5×1.225×22.1))
V_stall=√(1535.425122)
V_stall=39.18 m∕s or 141.10 km/h
Assuming that the runway distance is 1800 m, how do I calculate the minimum runway distance in order for the aircraft to take-off? What can I do to reduce the distance needed to take-off while keeping the final velocity at 39.18 m/s? Is there a way to increase the acceleration of the aircraft while keeping the same final velocity?

Here are some requirements that my aircraft must fulfill:

1. Maximum take-off weight of 4000 kg (39240 N)​
2. Take-off must take place at 1.25 times the stall speed, but less than 200 km/h​
3. Landing must take place at 1.3 times the stall speed, but less than 180 km/h​
4. Aircraft must take-off and land at a runway with a distance of 1800 m or less​

2. Nov 21, 2015

### SteamKing

Staff Emeritus
You're missing some key components of aircraft performance to make an intelligent study.

Chiefly, what sort of power or thrust can your aircraft's engines produce? Are they recips w/propellers, turboprops, turbojets, ramjets, what?

You can make a barn door fly if you attach enough power to it.

Aircraft performance is dependent on much more than just picking an airfoil for the wing.

Here is a bizjet which has roughly your max. TO weight:

https://en.wikipedia.org/wiki/Cessna_Citation_Mustang

3. Nov 21, 2015

### GeForce7

Thank you for the reply. Let's say my aircraft can produce a maximum take-off thrust of 14000 N, how can I work out the calculation? Can I use the F=ma formula to work out the acceleration? Do pilots always go for maximum engine thrust during take-off?

4. Nov 21, 2015

### SteamKing

Staff Emeritus
You pretty much have to. Of course, F is not just the thrust, it's Fnet, which includes the rolling friction on the landing gear, and the drag on the plane as it increases speed during the takeoff roll, all acting against the thrust of the engines.

That's what it's there for.

The aircraft on take-off typically has a max load: full fuel, full passengers, full cargo, whatever.

Say the airfield is at a higher elevation; there's no wind blowing down the runway; the air temperature is high, so the air density at altitude is relatively low; you have a short runway with height obstructions just beyond the end of the field. This is the time to crank the throttles wide open, cross your fingers, and hope that your take-off calculations are right, and that you don't have an engine malfunction halfway down the runway.

Once the aircraft is off the ground and flying, the engines are required only to produce enough thrust to counter the drag of the aircraft as it is flying at the set cruising speed and altitude.

5. Dec 13, 2015

### David Lewis

You can both increase acceleration and reduce take off distance by reducing weight. (I don't know if your assignment permits that.) Another way to reduce take off distance is increase maximum lift coefficient or wing area. But in any case you want to plug the total lift of the airplane into your formulae, which is normally wing lift minus stabilizer download, assuming a relatively conventional configuration.

Bear in mind that lift and weight are equal and opposite only in level flight. Climbing or descending, weight is equal and opposite to the vector sum of total aerodynamic force and thrust.

Last edited: Dec 13, 2015
6. Jan 6, 2016

### GeForce7

Thanks you guys for providing me the assistance. Right now, I need to calculate the thrust that can a turbofan engine can produce using F=ma equation. However, I'm not sure how I can approach it. Can I use a hypothetical thrust value and work out the calculation backward? For example, if my maximum take-off thrust is 14,000 N, how do I calculate the mass of the air and acceleration?

7. Jan 6, 2016

### SteamKing

Staff Emeritus
It's not clear what you want to do here.

Are you trying to figure out how fast an aircraft can move with a set max. thrust, or are you trying to design an engine to give a max. thrust value?

If the latter, it will take much more than F = ma to figure the performance of a turbofan engine.

https://www.grc.nasa.gov/www/k-12/airplane/turbfan.html

8. Jan 6, 2016

### GeForce7

You are correct. I'm trying to figure out an engine that can give me a maximum thrust value of 14,000 N. The formula from NASA's website is too difficult for me to figure it out myself. The thrust value is taken from Pratt and Whitney PW615F which is capable of generating a maximum take-off thrust of 6000 N. Since I'm clueless on the engine thrust calculation, I can only use the thrust value of an engine and pluck it into an equation to calculate the acceleration of my aircraft which is 1.49 m/s^2. In theory, my aircraft can lift off easily and only require a runway distance of 850.05 m which is significantly lesser than requirement. My only issue now is that I need to show the lecturer that the engine thrust value can be calculated.

9. Jan 6, 2016

### SteamKing

Staff Emeritus
P & W make more than one model of jet engine. There are other engine manufacturers out there, like GE, Rolls-Royce, etc. A thrust of 14000 N is about 3100 lbf, so maybe one of the makers of smaller engines, like Garrett or Williams, would be more what you are looking for. A Williams FJ44-4 is rated at 16000 N.

http://www.williams-int.com/products/fj44-4

Most of these performance data, like static thrust, fuel consumption, etc., are taken not from calculations, but actual engine tests after the design is finished.

10. Jan 6, 2016

### GeForce7

Ok. I will try to convince them that the engine thrust calculation is not feasible and I should just use the thrust value given by the manufacturer for better accuracy.

11. Jan 6, 2016

### cjl

It's pretty rare for them to use max thrust at takeoff actually - they calculate thrust required based on runway length, air density, takeoff weight, and whether or not there are any obstacles to clear (like a large hill or building just past the end of the runway), and then base their takeoff thrust on that. Full thrust is rarely used because it increases engine wear.

12. Jan 6, 2016

### David Lewis

You can calculate thrust using F=ma if you know engine power and the diameter of the engine's ideal (Rankine) actuator disk.

13. Jan 6, 2016

### cjl

Jet engines don't have a fixed power though (nor do they have a fixed thrust, though that's often a simplifying assumption that's made with them). It's rather complicated to actually get a decent estimate of the thrust of a jet engine.

14. Jan 7, 2016

### David Lewis

Good point. Thrust varies depending on speed even at a fixed throttle setting. At the beginning of the take off run, you have static thrust, but as the airplane picks up speed, thrust changes.

I believe the relationship between thrust and power is:
power = sq rt [(thrust cubed) divided by (2 * air density * area of actuator disk)]

15. Jan 7, 2016

### cjl

What kind of power are you trying to get? Power applied to the air? Power applied to the aircraft? The first is rather complicated for jet engines, as the core air has a pretty significant amount of power added in the form of heat, while the bypass air is pretty much just passing through a ducted fan. The second one is easy (but largely useless), as it will simply be the thrust multiplied by the aircraft velocity. I don't see how your formulation could be correct for any definition though - there should definitely be a velocity dependence somewhere in there..

16. Jan 7, 2016

### David Lewis

The formula only tells you how much power is consumed in producing a given amount of thrust. It doesn't matter how fast the airplane is going. If the airplane is not moving, for example, the engine still puts out power and still produces thrust.

17. Jan 7, 2016

### cjl

It does matter how fast the airplane is going though - the same amount of power will produce different amounts of thrust both based on the speed of the air traveling through the engine and the massflow of the air traveling through the engine. You really can't separate the two.

18. Jan 7, 2016

### Staff: Mentor

Those two don't work together.

19. Jan 8, 2016

### GeForce7

Hi guys, I would like to ask a question with regards to TSFC (Thrust Specific Fuel Consumption). The formula for calculating TSFC is fuel flow (kg/hour) divided by the thrust being generated by the engine at that point of time. Is it correct to assume that the fuel flow changes as you increase or decrease the thrust? If not, is the fuel flow constant? For example, the engine will burn less fuel (lower fuel flow) during cruising and burn more fuel (high fuel flow) during take-off?

20. Jan 8, 2016

### Staff: Mentor

Sure.

21. Jan 8, 2016

### jim hardy

Airklners use movable extensions to change the airfoil shape
ever watch out the window on approach to landing? The wing becomes much wider and curved.

22. Jan 8, 2016

### cjl

They're called flaps and slats, specifically, and they're important in allowing the airplane to have an airfoil shape that can provide low drag in cruise while still being able to generate enough lift to land at a fairly low speed.

23. Jan 8, 2016

### GeForce7

Using an example taken from Wikipedia (https://en.wikipedia.org/wiki/Williams_FJ44#Specification), I took the values from FJ-44-1A and did some calculation.

Formula (https://www.grc.nasa.gov/www/k-12/airplane/sfc.html):

TSFC = Fuel Consumption in lb per hour/Thrust in lbf
0.456= Fuel Consumption in lb per hour/1900 lbf
Fuel Consumption in lb per hours=0.456*1900
Fuel Consumption in lb per hours=866.4 lb/h OR (392.99 kg/h)

Assuming my calculation is correct, the TSFC given by the manufacturer is for maximum thrust? If so, the fuel consumption at maximum thrust is 866 lb per hour (392.99 kg per hours)? Is there a way for me to calculate the fuel consumption at a certain thrust value (for example, during low speed cruise)?

24. Jan 10, 2016

### David Lewis

The power (as a function of thrust) formula I gave you gives you induced power, if my understanding is correct. Therefore power must be divided by the efficiency of the propulsion system.

System efficiency is sometimes divided into two parts: How efficiently chemical energy in the fuel is converted to heat and mechanical energy, and then what portion of that energy is converted to thrust.

25. Jan 10, 2016

### cjl

Yes, TSFC is usually specified at full (non-afterburning) thrust. Unfortunately, there isn't an easy relationship to calculate fuel consumption at lower speeds - there are some calculations that can be done, but they usually end up rather complex (I'll dig out my propulsion textbook in a bit and see what it says).