AEROSPACE PROPULSION & AERODYNAMICS

AEROSPACE PROPULSION & AERODYNAMICS
Question 1
The propeller blades on the B29 are variable pitch, assume a geometrical pitch angle range of zero to alpha_pitch degrees and the blade twist ensures the pitch angle at the tip is alpha_twist degrees lower than at the spinning nose cone or hub. Also, assuming that each of the four propeller blades have lifting surfaces of 5 1/2 foot span, 6 inch chord and at the root of each blade is an axle embedded in the 1 foot diameter hub.
With constant reference to Fig. 88 from your notes.
Assume for a minute that the variable pitch propeller is replaced with a fixed pitch propeller, each blade has cross-sectional profile shaped based on NACA2412 and the geometric pitch angle is set at 15 degrees. As the aircraft accelerates on the runway at mean sea level altitude to the point of take off at a speed of 100mph, the engine is driving the propeller at maximum power turning at 2800 rpm.
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1a] Using circular motion equations, calculate the prop shaft angular velocity in rads/sec and blade tip speed in m/s.

1b] Calculate the resultant relative velocity at the blade root and tip, calculate the relative Mach number at the blade tip, comment on the flow speed regime towards the tip and the implications of the chosen blade aerofoil profile.

1c] The geometric pitch angle the blades are set at is given, using the forward speed of the aircraft as the axial velocity and calculated velocities for the blade root and tip from 1b draw a vector diagram to show the resultant relative velocity at both root and tip, calculate the helix angle at the blade root and tip so you can then calculate the effective angle of attack at root and tip.

Question 2
2a] Then replacing the constant pitch propeller with another fixed propeller with applied twist to the propeller blades, again whilst the aircraft is on the verge of take off on a runway at msl, if the effective angle of attack locally along the entire propeller blade span is 2 degrees calculate the geometric twist angle alpha_twist where
alpha_twist = alpha_tip – alpha_root
(Hint: recall helix angles are the same as 1c, blade pitch angle = effective AoA + helix angle)

2b] Assuming that the low speed lift and drag curves hold for all local velocities incident on the propeller blades, taking values for CL and CD from the literature for the NACA2412 blade, calculate the lift and drag per unit span on a propeller blade at root and tip using the resultant relative velocity calculations from 1b.
(Hint: lift per unit span L/b = 0.5 x rho x V^2 x c x CL, where c is the chord and b is the blade span)
Again with reference to fig. 88 in the notes, recall that lift is perpendicular and drag parallel to the relative velocity vector line, with that in mind draw a diagram to show how the aerodynamic forces should be resolved to get thrust and calculate the thrust force at the blade tip, then plot a graph of local thrust against diameter for one propeller blade modelling the local thrust (per unit span) curve using the following equation,
local thrust = 0.25 x tip thrust x r^2
Using the Trapezium rule (as in your lab report) to represent the area under your graph estimate the thrust generated by a single propeller blade, then calculate the thrust from the
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propeller i.e. from all of the blades and also from the entire aircraft i.e. adding up to account for all propellers.

2c] using the calculated thrust from all propellers, calculate the power generated by the aircraft using
power = thrust x take off speed
If the engine power of a single B29 powerplant is 2200hp calculate the propeller efficiency as a ratio of thrust to engine power.

Question 3
3a] Changing propeller again back to the standard specification variable pitch propeller from a B29, assume the twist angle is set up perfectly for cruise such that the local AoA is 2 degrees, firstly calculate the density of air at the cruise altitude of 31850 feet, state the cruise velocity 220mph in metres.

3b] Assuming a more efficient (propeller specific) aerofoil cross-section is chosen for the blades such that the propellers can achieve 90% efficiency, the B29 would fly above the ceiling height at 47000 feet during missions with the engines at maximum thrust, in order for the aircraft to achieve a record breaking speed of 410.43mph (for a piston engine driven aircraft over long distance) calculate the max thrust power drawn from the 2200hp engines and the max thrust available using simple efficiency equations.
If the aircraft is not accelerating then thrust is equal to drag, therefore using the following equation which describes the thrust required curve
D = Dmin + m (V – Vcruise)^2 [D Drag, V Velocity, m constant]
where the constant m = 10/3 and Vcruise for the B29 is known, firstly find Dmin which can be assumed to occur at the cruise condition using
Dmin = 0.5 x rho x Vcruise^2 x S x cd
[Hint: use your calculated value for air density at the cruise speed altitude from 3a, the wing area of the B29 is 161m^2, use an aircraft drag coefficient of 0.024]
Then assuming max thrust available is constant for all velocities, also using the equation describing thrust required above plot the thrust available and thrust required on the same graph against airspeed, mark on the graph Vmin, Vroc & Vmax assuming the stall speed is 100mph at msl, comment on how the actual graphs for the B29 might look different and why.

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Question 4
4a] Again thinking about the B29 on take-off, with reference to the force body diagram Fig. 76 in the aircraft manoeuvres section of the notes show that at the point of leaving the msl runway on takeoff the acceleration is given by
a = (T – D) / m
If the thrust during take off is constant and at the maximum output from 80% efficient 2200hp engines calculate the total thrust from all powerplants combined using the propeller efficiency equation, if the weight of the aircraft is 100000lbs and the lift coefficient for the high lift configuration is 2.25 calculate the drag at the point of lift off if L/D is 16.8 and assuming the take off speed is 100mph as before, therefore now calculate the acceleration of the aircraft at the point of lift off, then show that the Lift to Weight ratio at the point of lift off is approximately equal to 1 as should be expected.
If the wheel coefficient of friction mu is 0.15 calculate the acceleration at the point where the aircraft is travelling at half the take off speed, then in order for us to get a simple estimation for the take off distance if we assume that the acceleration was constant right from standstill (chocks away) to lift off, use the acceleration value at lift off to calculate the duration of the acceleration phase of take-off on the runway and also the take off distance, the take off distance of the B29 was around 5000feet discuss how your value compares and offer arguments as to why it differs with reference to the acceleration equation in the notes.

4b] If the acceleration then drops off to zero but the 80% efficient engines still run at max thrust and the aircraft climbs at a steady speed of 100mph in the high lift wing configuration, state the lift and drag of the aircraft, then calculate the climb angle and load factor assuming msl air density, then find the rate of climb (roc), does your roc value seem reasonable compared to aircraft of a similar size.

4c] The aircraft levels out at 31850feet flying at the highest angle of attack possible to maintain straight and level flight without stalling, the lift coefficient in this configuration is a very high number 4, recall and state the air density at this altitude and calculate the velocity necessary to support the aircrafts weight, assume now then that your calculated value is the stall speed at the cruise altitude, if the aircraft banks into a turn port wing down calculate the load factor for a bank angle of 30 degrees and the stall velocity in the turn, then calculate the turn radius r and work out the time it would take to execute a 90 degree turn using standard circular motion equations.
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