Aeronautics

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Low BPR turbofan/jet pre-design
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■1. Which sort of power generator?
> Piston engine will be too much inertia to accelerate supersonic in a short time
> Piston engine will save fuel consumption, but its big size, then wider surface, will significantly impact the aircraft drag &
mass, implying more engine power
 In fine as much fuel burn as gas generator as FB = SFC x PW
> Electrical engine, with batteries, in such a hot environment as afterburning jet engine shall not be the best reliability option
> The most intuitive choice is the gas generator
 Small sizing
 Low mass & inertia for good acceleration
■2. Fan thermodynamics
> Assume Mach 0.8 flight at sea level
> Assume some quite high fan pressure ratio at 2.0
> P2/T2 are also gas generator inlet conditions
Station Pt (Pa) Tt (K)
Ambiant Pamb/Tamb 100000 300 1bar, 27°C
Fan inlet P1/T1 142961 332 Pt=Pamb + 1/2 rho V² , Tt/Tamb=(Pt/Pamb)^((gamma-1)/gamma)
Fan outlet P2/T2 285922 413 FPR=2, if adiabatic : T2ad/T1=FPR^(0.4/1.4) , as efficiency 0.9 : (T2-T1) = (T2ad-T1)/0.9
Nozzle Ve (m/s) = 571 Ve = sqrt((P2-Pamb)/0.5rho), to determine rho : Ps=Pamb, Ts/T2=(Pamb/P2)^(0.4/1.4)

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Low BPR turbofan/jet pre-design
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■3. Gas generator thermodynamics
> Gas generator first part is compressor
> Assuming ~25 OPR (overall pressure ration = Pcombustion/P1)
 GG compressor shall have a 9 pressure ratio
> Choose the inlet turbine temperature
 1450K enables uncooled turbine rotor
 Cooled one could reach 1700K to 1900K
> Fuel/Air ratio : FAR = [ Cp_air x (T4-T3) ] / FHV ~ 2% << 14%
 => there is still enough oxygen for afterburning
■4. Fan turbine & primary nozzle
> Here we have to choose BPR to determine the fan turbine power.
 T5-T6 = (BPR+1) x (T2-T1) / turbine_efficiency
> I will assume a 1 BPR
Station Pt (Pa) Tt (K)
GG inlet P2/T2 285922 413
Combustion inlet P3/T3 2573298 960 0.85 efficiency : (T2-T1) = (T2ad-T1)/0.85
Combustion outlet P4/T4 2521832 1800 T4 = 1450K for uncooled turbine (1700 to 1850K for cooled turbines) ; 2% pressure drop
HP Turbine outlet T5/P5 709704 1192 0.9 efficiency : (T4-T5) = (T3-T2)/0.9 ; T5ad = T4-0.9*(T4-T5) = T4-T3+T2
HP Turbine outlet T5/P5 709704 1192 0.9 efficiency : (T4-T5) = (T3-T2)/0.9 ; T5ad = T4-0.9*(T4-T5) = T4-T3+T2
LP turbine outlet T6/P6 426104 1013 0.9 efficiency : (T5-T6) = (T2-T1).(1+1)/0.9
Cold nozzle Ve (m/s) = 1119 Ve = sqrt((P6-Pamb)/0.5rho), to determine rho : Ps=Pamb, Ts/T6=(Pamb/P6)^(0.4/1.4)

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Low BPR turbofan/jet pre-design
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■Specific thrust of this engine, M0.8 no afterburning
T/q = 1 / (1+BPR) x Ve_afterburning + BPR / (1+BPR) x Ve_fan – V0
T/q = 559.5 + 285.5 – 272 = 573 N/kg
Eta_prop = 2 x T x V / [ BPR/(BPR+1) Ve_fan² + 1/(BPR+1) VE_nozzle² -V0² ]
Eta_prop = 2 x T x 0.8 x 340 / [ 0.5 x 1119² + 0.5 x 571² - 272² ] ~ 44%
Station Pt (Pa) Tt (K)
Ambiant Pamb/Tamb
100000 300
Fan inlet P1/T1 142961 332
Fan outlet P2/T2 285922 413
Nozzle Ve (m/s) = 571
Station Pt (Pa) Tt (K)
GG inlet P2/T2 285922 413
Combustion inlet P3/T3
2573298 960
Combustion outlet P4/T4
2521832 1800
HP Turbine outlet T5/P5
709704 1192
LP turbine outlet T6/P6
426104 1013
Cold nozzle Ve (m/s) = 1119
cold 573
44%
M0.8
T/q (kN/kg)
Etaprop

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Turboprop pre-design
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■1. Sizing the propeller
2 formulas for Propeller choice :
> J = V0/(N.D) – speed coefficient
 V0 = airplane speed (m/s)
 N = propeller rotation speed (rev/s)
 D = diameter (m)
> Ct = F/(rho.N².D4)
 F = propeller traction (N)
 Rho = air density (kg/m³)
 N = propeller rotation speed (rev/s)
 D = diameter (m)
2 specifications :
> Thrust
> Airplane speed
at key points, as cruise & take-off

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Turboprop pre-design
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■2. Find some propeller maps
Ct, Efficiency (h) characteristics function of J factor and propeller pitch

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Turboprop pre-design
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■3. Specification
As example (pure imagination), let’s take
> Cruise is 1kN traction at 100m/s (~200knts) @ 4500m, ISA
> Take-off is 5kN traction at 70m/s (~140knts) @ 0m, ISA
■4. Propeller diameter first guess
Choose propeller diameter
>D = V0/(N.J)
=> for a typical 2000RPM propeller, Diameter is here 3/J meters
>Let’s start with a 2m propeller, J=1.5
>Best efficiency is for 35° pitch,and then Ct = 0.06

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Turboprop pre-design
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■5. Propeller cruise pitch
Get necessary traction coefficient at cruise
for choosen J ratio
>Here, J = 1.5 => D = 2m
>Ct = F/(rho.N².D4) = 0.072
F = 1kN, D = 2m, N = 2000/60 rev/s
Rho = P0/(Rair.T0) = 0.78kg/m³
○ Rair = 287.05
○ P0 = 57.7kPa @ 4500m, ISA
○ T0 = 258.5K @4500m, ISA
>Pitch ~36°
■6. Control propeller efficiency
From Ct & J factors,
look for propeller efficiency for the dedicated pitch value
>Here, h ~ hmax
>J = 1.5 was a good choice for cruise operating point

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Turboprop pre-design
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■7. Propeller take-off pitch
Get necessary traction coefficient at take-off
>J = V/(N.D) = 1.1
V = 70m/s, D = 2m, N = 2000/60 rev/s
>Ct = F/(rho.N².D4) = 0.230
F = 5kN
Rho = P0/(Rair.T0) = 1.23kg/m³
○ Rair = 287.05
○ P0 = 101.3kPa @ 0m, ISA
○ T0 = 288.2K @ 0m, ISA
>! Ct > Ct max (~0.18) !
Speed or diameter is too low

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Turboprop pre-design
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■8. Adapt take-off rotation speed
A safer traction coefficient would be Ct = 0.15
>N² = F/(rho.Ct.D4) = 1693 rev/s
>=> N ~ 2470 RPM
We can increase take-off propeller speed by 25% vs. cruise
to reach the traction specification
>J factor value is now 0.84, & pitch is 32°
But efficiency is poor (~0.65)
To reach better efficiency at take-off, we need
>To decrease the pitch
>Or to increase the J factor

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Turboprop pre-design
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■9. Iterate on propeller sizing
To decrease take-off pitch, we need to decrease J factor
>Increase propeller diameter
Proportional to 1/J
>In the same time, it will decrease the needed Ct
Proportionnal to J4
Change cruise J factor from 1.5 to 1.4
(diameter reaches 2.1m)
>Cuise Ct coefficient changes from 0.72 to 0.055
>Take-off J factor moves from 0.84 to 0.78
>And associated Ct coeff. from 0.15 to 0.11
Resulting cruise pitch is 32° with still >.8 efficiency
And take-off pitch is now 25° only
and efficiency reaches 0.75

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Turboprop pre-design
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■10. Define engine power needed
Look for power coefficient for both cruise and take-off
Read the diagram using known J & pitch values from 9.
>Cruise : J = 1.4 & pitch = 32°
Cp value is ~0.09
>Take-off : J = 0.78 & pitch = 25°
Cp value is ~0.11
Then you get the propeller power thanks to the formula
PW = Cp.rho.N³.D6
Cruise power = 55kW
Take-off power = 206kW

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Turboprop pre-design
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■11. Which operating point is engine sizing point?
Available engine power is varying with altitude
>To compare all altitude powers we correct to 0m ISA conditions
𝑃𝑊
𝑐𝑜𝑟𝑟 =
𝑃𝑊 . 101325𝑃𝑎 . 288.15𝐾
𝑃0 . 𝑇0
In this example
>Take-off power of 206kW @ 0m, ISA leads to Pwcorr = 206kW
>Cruise power of 55kW @ 4500m, ISA leads to Pwcorr = 101kW
>Take-off power of 206kW is sizing the engine

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Turboprop pre-design
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■12. Last, find the engine !
206kW engine (276 horse power) is needed by the propeller
>Don’t forget to consider ~10% installation losses on the engine
>Looking for 300shp engine
For such power, solution will probably be turbine engine
But some piston engine would still be available
Seems one solution exists 

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Turboprop pre-design
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■13. And very last, find the propeller !
Available manufacturers are depending the size & power
>For such power, solution could be found by MT-Propellers or Hartzell for example
>You can check your pre-design
2700RPM @ 350 HP
○ We got 2500 for 300 HP
○ Not so bad 
78”” diameter ~1.98m
○ We got 2.1m
○ Not so bad, still 
Here is a possible solution 