Weight Estimation Aircraft weight, and its accurate prediction, is critical as it affects all aspects of performance. Designer must keep weight to a minimum as far as practically possible. Preliminary estimates possible for take-off weight, empty weight and fuel weight using basic requirement, specification (assumed mission profile) and initial configuration selection.
1. Take-off weight (WTO) – (Roskam method) Note that other methods (e.g. Raymer) use slightly different terminology but same principles. WTO = WOE + WF + WPL Where: WOE (or WOWE ) = operating weight empty WF = fuel weight WPL = payload weight Operating weight empty may be further broken down into: WOE = WE + Wtfo + Wcrew Where:
WE = empty weight Wtfo = trapped (unusable) fuel weight Wcrew = crew weight Empty weight sometimes further broken down into: WE = WME + WFEQ Where: WME = manufacturer’s empty weight WFEQ = fixed equipment weight (includes avionics, radar, air-conditioning, APU, etc.)
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2. Preliminary Weight Estimation - Overview • • •
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All textbooks use similar methods whereby comparisons made with existing aircraft. In Roskam (Vol.1, p.19-30), aircraft classified into one of 12 types and empirical relationship found for log WE against log WTO. Categories are: – (1) homebuilt props, (2) single-engine props, (3) twin-engine props, (4) agricultural, (5) business jets, (6) regional turboprops, (7) transport jets, (8) military trainers, (9) fighters, (10) military patrol, bombers & transports, (11) flying boats, (12) supersonic cruise. Most aircraft of reasonably conventional design can be assumed to fit into one of the 12 categories. New correlations may be made for new categories (e.g. UAVs). Account may also be made for effects of modern technology (e.g. new materials) – method presented in Roskam Vol.1, p.18. Raymer method uses Table 3.1 & Fig 3.1 (p.13).
Preliminary Weight Estimation Process • • • • • •
Process begins with guess of take-off weight. Payload weight determined from specification. Fuel required to complete specified mission then calculated as fraction of guessed take-off weight. Tentative value of empty weight then found using: WE(tent) = WTO(guess) – WPL - Wcrew - WF - Wtfo Values of WTO and WE compared with appropriate correlation graph. Improved guesses then made and process iterated until convergence. 2
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Note that convergence will not occur if specification is too demanding
Initial Guess of Take-Off Weight • •
Good starting point is to use existing aircraft with similar role and payload-range capability. An accurate initial guess will accelerate the iteration process.
Payload Weight & Crew •
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WPL is generally given in the specification and will be made up of: – passengers & baggage; cargo; military loads (e.g. ammunition, bombs, missiles, external stores, etc.). Typical values given in Roskam Vol.1 p8. Specific values for some items (e.g. weapons) may be found elsewhere.
Mission Fuel Weight • •
This is the sum of the fuel used and the reserve fuel. WF = WF(used) + WF(res) Calculated by ‘fuel fraction’ method. – compares aircraft weights at start and end of particular mission phases. – difference is fuel used during that phase (assuming no payload drop). – overall fraction is product of individual phase fractions.
Simple Cruise Mission Example 1. 2. 3. 4. 5.
start & warm-up taxi take-off climb cruise 3
6. loiter 7. descend 8. taxi • •
Fuel fractions for fuel-intensive phases (e.g. 4, 5 & 6 above) calculated analytically. Non fuel-intensive fuel fractions based on experience and obtained from Roskam (Vol 1, p12), Raymer, etc.
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Using Roskam’s method/data for a transport jet (Vol.1, Table 2.1):
Phase 1 (start & warm-up) W1/WTO = 0.99. Phase 2 (taxi) W2/W1 = 0.99. Phase 3 (take-off) W3/W2 = 0.995. Phase 4 (climb) For piston-prop a/c: 1 Ecl = 375 Vcl
ηp cp
L W ln 3 D cl W4 cl
For jet a/c: 1 Ecl = cj
L W ln 3 cl D cl W4
where: Ecl = climb time (hrs), L/D = lift/drag ratio, cj is sfc for jet a/c (lb/hr/lb), cp is sfc for prop a/c (lb/hr/hp), Vcl = climb speed (mph), ηp = prop efficiency, W3 & W4 = a/c weight at start and end of climb phase. • •
Initial estimates of L/D, cj or cp, ηp and Vcl made from Roskam or Raymer databases for appropriate a/c category. Alternatively, use approximations, e.g. from Roskam Vol.1, Table 2.1 (W4/W3=0.98 for jet transport, 0.96 to 0.9 for fighters).
Phase 5 (cruise) • Weight fraction calculated using Breguet range equations. • For prop a/c:
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For jet a/c: V Rcr = cj
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L W ln 4 D cl W 5 cr
These give the range in miles. 4
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For jet a/c, range maximised by flying at 1.32 x minimum drag speed and minimising sfc. – wing operates at about 86.7% of maximum L/D value. – cruise-climbing can also extend range. • For prop a/c, range maximised by flying at minimum drag speed and sfc. – wing operates at maximum L/D value. Initial Estimates of Lift/Drag Ratio (L/D) Using Roskam (Table 2.2 – selected values
Homebuilt & single-engine Business jets Regional turboprops Transport jets Military trainers Fighters Military patrol, bombers & transports Supersonic cruise
cruise 8 - 10 10 – 12 11 – 13 13 – 15 8 – 10 4–7 13 – 15 4-6
loiter 10 - 12 12 - 14 14 – 16 14 - 18 10 - 14 6–9 14 – 18 7–9
Specific Fuel Consumption Initial estimates of cj (lb/hr/lb) • Using Raymer (Table 3.3):
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cruise
loiter
Turbojet
0.9
0.8
Low-bypass turbofan
0.8
0.7
High-bypass turbofan
0.5
0.4
Roskam Vol.1 Table 2.2 (p.14) gives a/c category-specific values
Jet aircraft - Initial estimates of cj (lb/hr/lb) cruise
Loiter
Business & transport jets
0.5 - 0.9
0.4 - 0.6
Military trainers
0.5 - 1.0
0.4 - 0.6
Fighters
0.6 - 1.4
0.6 - 0.8
Military patrol, bombers, transports, flying boats
0.5 – 0.9
0.4 - 0.6
Supersonic cruise
0.7 – 1.5
0.6 - 0.8
Prop aircraft - Initial estimates of cp (lb/hr/hp) • Using Raymer (Table 3.4): cruise 5
loiter
Piston-prop (fixed pitch)
0.4
0.5
Piston-prop (variable pitch)
0.4
0.5
Turboprop
0.5
0.6
• Take propeller efficiency (ηp) as 0.8 or 0.7 for fixed-pitch piston-prop in loiter. Prop aircraft - Initial estimates of cp (lb/hr/hp) & ηp • Using Roskam (Table 2.2): Single engine Twin engine Regional turboprops Military trainers Fighters Military patrol, bombers & transports Flying boats, amphibious
Cruise 0.5 – 0.7, 0.8 0.5 – 0.7, 0.82 0.4 – 0.6, 0.85 0.4 – 0.6, 0.82 0.5 – 0.7, 0.82 0.4 – 0.7, 0.82 0.5 – 0.7, 0.82
loiter 0.5 – 0.7, 0.7 0.5 – 0.7, 0.72 0.5 – 0.7, 0.77 0.4 – 0.6, 0.77 0.5 – 0.7, 0.77 0.5 – 0.7, 0.77 0.5 – 0.7, 0.77
Phase 6 (loiter) • Fuel fraction (W6/W5) found from appropriate endurance equation as in Phase 4. • For jet a/c, best loiter at minimum drag speed (maximum L/D value); for prop a/c at minimum power speed. Phase 7 (descent) W7/W6 = 0.99. Phase 8 (taxi) W8/W7 = 0.992. Overall Fuel Fraction (Mff) W W W W W W W W M ff = 8 7 6 5 4 3 2 1 W7 W6 W5 W4 W3 W2 W1 WTO • Mission fuel used (WF(used)) WF (used ) = (1 − M ff ) WTO = • • • • •
WF then found from equation (5), by adding reserve fuel (WF,res). This then allows for tentative value for WE(tent) to be found, from eq. (4). This may be plotted with WTO on appropriate a/c category graph to check agreement with fit. If not, then process must be iterated until satisfactory. Two other possible mission phases may need to be considered for certain aircraft: • manoeuvring • payload drop Manoeuvring Fuel • Breguet range equation may be used with range covered in turn (R turn) from perimeter length of a turn (P turn) multiplied by number of turns (N turn).
Rturn = N turn Pturn
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For manoeuvre under load factor of n: V2 Pturn = 2π g n 2 − 1
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Treated as separate sortie phase with change in total weight but no fuel change. 6
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Fuel fraction taken as 1 but subsequent phases corrected to allow for payload drop weight change. Roskam Vol.1 pp.63-64 gives details. e.g. if W5 and W6 are weights before and after payload drops: W5 =
W5 W4 W3 W2 W1 WTO W4 W3 W2 W1 WTO
W6 = W5 − WPL
Worked Example – Jet Transport Specification Payload: Crew: Range: Altitude: Cruise speed: Climb: Take-off & landing: • • •
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150 passengers at 175 lbs each & 30 lbs baggage each. 2 pilots and 3 cabin attendants at 175 lbs each and 30 lbs baggage each. 1500 nm, followed by 1 hour loiter, followed by 100 nm flight to alternate and descent. 35,000 ft for design range. M = 0.82 at 35,000 ft. direct climb to 35,000 ft at max WTO. FAR 25 field-length of 5,000 ft.
WPL = 150 x (175 + 30) = 30,750 lbs Wcrew = 1,025 lbs Initial guess of WTO required, so compare with similar aircraft: – Boeing 737-300 has range of 1620 nm for payload mass of 35,000 lbs – WTO = 135,000 lbs. – Initial guess of 127,000 lbs seems reasonable. Now need to determine a value for WF, using the fuel fraction method outlined above. As in earlier example, for a transport jet:
Phase 1 (start & warm-up) W1/WTO = 0.99. Phase 2 (taxi) W2/W1 = 0.99. Phase 3 (take-off) W3/W2 = 0.995. 7
Phase 4 (climb) W4/W3 = 0.98. The climb phase should also be given credit in the range calculation. Assuming a typical climb rate of 2500 ft/min at a speed at 275 kts then it takes 14 minutes to climb to 35,000 ft. • Range covered in this time is approximately (14/60) x 275 = 64 nm. Phase 5 (cruise) • Cruise Mach number of 0.82 at altitude of 35,000 ft equates to cruise speed of 473 kts. • Using eq. (7b):
V Rcr = cj •
L W ln 4 cr D cl W5
Assumptions of L/D = 16 and cj = 0.5 lb/hr/lb with a range of 1500 – 64 (=1436 nm) yield a value of: W5/W4 = 0.909
Phase 6 (loiter) • Using eq. (6b):
1 L W Ecl = ln 3 c D cl W4 j cl • Assumptions of L/D = 18 and cj = 0.6 lb/hr/lb. • No range credit assumed for loiter phase. • Substitution of data into eq. (6b) yields: W6/W5 = 0.967 Phase 7 (descent) • No credit given for range. W7/W6 = 0.99. Phase 8 (fly to alternate & descend) • May be found using eq. (6b) again. • Cruise will now take place at lower speed and altitude than optimum – assume cruise speed of 250 kts (FAR 25), L/D of 10 and cj of 0.9 lb/hr/lb. • Gives: W8/W7 = 0.965 Phase 9 (landing, taxi & shutdown) • No credit given for range. W9/W8 = 0.992. Overall mission fuel fraction (Mff) • found from eq. (8) (with additional term for W9/W8) = 0.992x0.965x0.99x0.967x0.909x0.98x0.995x0.99x0.99 = 0.796 • Using eq. (9), WF = 0.204 WTO = 25,908 lbs Phase 9 (landing, taxi & shutdown) 8
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Using eq. (4): WE(tent) = WTO(guess) – WPL - Wcrew - WF – Wtfo ∴ WE(tent) = 127,000 – 30,750 – 1,025 – 25,908 - 0 = 69,317 lbs By comparing with Roskam Vol. 1, Fig. 2.9, it is seen that there is a good match for these values of WE and WTO, hence a satisfactory solution has been reached.
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