Chapter 3 Weight estimation - 2 Lecture 7 Topics

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1 Chater-3 Chater 3 eight estimation - 2 ecture 7 Toics 3.5 Estimation o uel raction ( / 0 ) Mission roile eight ractions or various segments o mission Fuel raction or warm u, taxing and take-o( 1 / 0 ) Fuel raction or climb ( 2 / 1 ) Fuel raction during cruise outline o aroach Fuel raction during loiter outline o aroach Estimation o (/D) max outline o aroach Drag olar o a tyical high subsonic jet airlane 3.5 Estimation o uel raction ( / 0 ) The weight o uel needed deends on the ollowing. I.Fuel required or mission. II.Fuel required as reserve. III.Traed uel which cannot be umed out. The uel required or the mission deends on the ollowing actors. a) Mission to be lown. b) Aerodynamics o the airlane viz. ( / D) ratio. c) SFC o the engine Mission roile a) Simle mission: For a transort airlane the mission roile would generally consist o (a) warm u and take o, (b) climb, (c) cruise, (d) descent, (e) loiter and () landing(fig.3.6). Sometimes the airlane may be required to go to alternate airort i the ermission to land is reused. Allowance also has to be made or head winds encountered en-route. Det. o Aerosace Engg., Indian Institute o Technology, Madras 1

2 Chater-3 Remarks: (i) For a military airlane the light roile could consist o (a) warm u and takeo, (b) climb, (c) cruise to target area, (d) erorming mission in the target area, (e) cruise back towards the base, () descent, (g) loiter and (h) land. In the target area the airlane may carry out reconnaissance, or dro bombs or engage in combat. Fig.3.6 Tyical mission roile o a transort airlane As additional examles o the mission roiles the ollowing three cases can be cited. (a) A trainer airlane, ater reaching the seciied area, may erorm various maneuvers and return to the base. (b) An airlane on a humanitarian mission may go to the desired destination, dro ood and relie sulies and return to the base. (c) In some advanced countries the doctors rom cities ly to the remote areas, examine the atients and ly back. ii) The various segments o the mission can be groued into the ollowing ive categories. (a) arm u, taxing and take-o. (b) Climb to cruise altitude. (c) Cruise according to a seciied light lan. This item is covered under the toic o Range in Perormance analysis. (d) oiter over a certain area or a seciied eriod o time. This item is covered under the toic Endurance in Perormance analysis. (e) Descent and landing. Det. o Aerosace Engg., Indian Institute o Technology, Madras 2

3 Chater eight ractions or various segments o mission The uel required in a articular hase o the mission deends on (a) the weight o the airlane at the start o that hase and (b) the distance covered or the duration o time or the hase. Keeing these in view, the aroach to estimate uel raction or chosen mission roile is, as ollows. i) et the mission consist o n hases. ii) The uel ractions or the hase i is denoted as i / i-1. iii) et 0 be the weight at the start o the light (say warm u) and n be the weight at the end o last hase (say landing). Then, n / o is exressed as: =... n 1 2 n-1 n n-2 n-1 iv)the uel ractions ( i / i-1 ) or all hases are estimated and ( n / 0 ) is calculated rom Eq.(3.26). Subsequently, the uel raction ( / 0 ) is deduced as: =K 1- n t 0 0 (3.26) ; where K t is actor allowing or traed uel. (3.26a) Fuel raction or warm u, taxing and take-o ( 1 / 0 ) Reerence 1.15, chater 7 and Re.1.18, chater 3 give rough guidelines or uel consumed during these hases o light. Reerence 1.12, Vol.I chater 2 gives breaku o uel used in warm-u, taxing and take-o or dierent tyes o airlanes. Based on this data, the rough guidelines are as ollows. For home built and single engined iston airlanes 1 / 0 is For twin engined turboros, jet transorts (both civil and military), lying boats and suersonic airlanes 1 / 0 is For military trainers and ighters 1 / 0 is Fuel raction or climb ( 2 / 1 ) The ollowing guidelines are given based on the data in Re.1.12,Vol.I, chater 2. The low seed airlanes including the twin-engined airlanes and lying boat cruise at moderate altitude (say 4 to 6 km) and hence 2 / 1 is taken as The military and civil transort jets cruise around 11 km altitude and 2 / 1 is Det. o Aerosace Engg., Indian Institute o Technology, Madras 3

4 Chater-3 taken as The ighter airlanes have very owerul engines and attain suersonic Mach number at the end o the climb. In this case, 2 / 1 is between 0.9 to Similarly, the suersonic transort airlanes which cruise at high altitudes (15 to 18 km), 2 / 1 is around 0.9. Reerence 1.15, chater 7 gives more elaborate rocedure to estimate 2 / 1 and is ollowed or ighter and suersonic cruise airlanes Fuel raction during cruise outline o aroach Equations (3.8) and (3.10) resent the Breguet ormulae or range o airlanes with engine-roeller combination and with jet engine resectively. Consult books on erormance analysis (e.g. section o Re.3.3) or the derivation o these equations. However, it may be ointed out that while deriving these ormulae it is assumed that the ollowing quantities remain constant during the light. (a) it coeicient. (b) Seciic uel consumtion (BSFC or TSFC). (c) Proeller eiciency or airlanes with engine-roeller combination and (d) Flight altitude. Equations or range can also be derived when the light velocity remains constant instead o the lit coeicient. Reerences 1.15, chater 7 and 1.18, chater 17 consider a slightly dierent but simler aroach. The derivation is as ollows. In a light at velocity V (in m/s), the distance dr (in km) covered when a quantity o uel d (in N) is consumed in time dt, is given as : dr = d x (km / N o uel) (3.27) Now, in a time interval dt, the distance covered in km is 3.6 V dt, where V is the light seed in m/s; the actor 3.6 is to convert velocity to kmh. Note dt is in hrs. Further, or jet engined airlanes the uel consumed, d, in the time interval dt is : d = TSFC x T x dt where T is in N, TSFC is N/N-hr or hr -1 and dt is in hrs. Det. o Aerosace Engg., Indian Institute o Technology, Madras 4

5 Chater-3 Hence, (km / N o uel) = 3.6V dt TFSC T dt Substituting this in Eq. (3.27) gives : 3.6V dt 3.6 V dr = d = d TFSC T dt TFSC T (3.28) Noting that, D T= = C C (/D) and d = - d, gives : -3.6 V (/D) dr = d (3.29) TFSC Assuming V, TSFC and (/D) to be constant and taking i-1 and i as the weights o the airlane at the beginning and the end o the cruise, and integrating Eq.(3.29), yields : -3.6 V (/D) R= ln TFSC i-1 i (3.30) i Or i-1 -R TSFC =ex ; V in m/s. (3.31) 3.6 V (/D) For an airlane with engine-roeller combination, the uel consumed in the time interval dt is : THP BSFC T V dt BSFC BHP dt = BSFC dt = η η 1000 Hence, (km / N o uel ) = 3.6 V dt TV BSFC dt η 1000 Substituting this in Eq.(3.27) yields: 3.6 V dt d TV BSFC dt η 1000 dr = = d Noting that, C 3600 η BSFC T D T= = andd =-d C /D η /D d dr = BSFC yields: (3.32) (3.33) Det. o Aerosace Engg., Indian Institute o Technology, Madras 5

6 Chater-3 Assuming η, BSFC and /D to be constant and integrating Eq.(3.33) gives: R = 3600 η i (/D) ln BSFC i-1 Or i Remarks: -R BSFC =ex 3600η (/D) i-1 (3.34) i) hile deriving Eq.(3.30) both V and C are assumed to be constant, during cruise. However, weight o the airlane decreases during the light and to satisy = = ()ρ V 2 SC the values o ρ should decrease as weight decreases. The consequence is, the altitude o the airlane should increase as the light rogresses. This is called Cruise climb. However, the change in the altitude is small and the light can be regarded as level light. ii) To evaluate the uel raction using Eq.(3.31) requires values o TSFC, V and (/D). hen Eq.(3.34) is used, the values o BSFC, (/D) and η are needed. iii) The ollowing may be ointed out. (a) For airlanes with engine-roeller combination, Eq.(3.34) shows that the uel required would be minimum when the light takes lace at a C corresonding to (/D) max. (b) For jet engined airlanes, Eq.(3.31) shows that or minimizing the uel required, the roduct V(C /C D ) should be maximum. Since, V is roortional to 1/( C ( C ), the quantity (VC /C D ) is maximised when C corresonds to /C D ) max. Assuming arabolic olar i.e. C D = C D0 + KC 2, it is shown that the value o C corresonding to ( C /C D ) max is C DO /3K. The value o (/D) or this value o C is (/D) max Fuel raction during loiter outline o aroach Equations (3.9) and (3.11) resent the Breguet ormulae or endurance o airlanes with engine-roeller combination and with jet engine, resectively. Books on erormance analysis (e.g. section o Re.3.3) be consulted or Det. o Aerosace Engg., Indian Institute o Technology, Madras 6

7 Chater-3 derivation o these equations. As mentioned earlier, the derivations o these ormulae assume that the ollowing quantities remain constant during the light. (a) it coeicient (b) Seciic uel consumtion (BSFC or TSFC) (c) Proeller eiciency or airlanes with engine-roeller combination and (d) Flight altitude. Equation or endurance can also be derived when the light velocity remains constant instead o the lit coeicient. Reerence 1.18 chater 3 considers a slightly dierent but simler aroach. The derivation is as ollows. In a light at velocity V, the time elase de (in hr) when a quantity o uel d (in N) is consumed is given by : hr d de = d = N o uel N o uel / hr. For a jet engined airlane, (N o uel/ hr ) = TSFC x T d Hence, de = TSFC T Noting that, T = (C D /C ) = /(/D) and d = - d gives: /D de = - d (3.35) TSFC et (a) (/D) and TSFC be assumed to remain constant during the light, (b) i-1 and i the weights o the airlane at the beginning and end o light. On integrating Eq.(3.35), gives /D E= ln TSFC i-1 i (3.36) i Or i-1 -E TSFC =ex (3.37) /D For an airlane with engine-roeller combination, the quantity (N o uel/ hr) is : T V (N o uel/ hr) = BSFC BHP = BSFC 1000 η Det. o Aerosace Engg., Indian Institute o Technology, Madras 7

8 Chater-3 Consequently, d de = = TV CD BSFC BSFC V 1000η d 1000 η C = η (/D)d BSFC V (3.38) Assuming Eq.(3.38) yields: E = Or η, BSFC, (/D) and V to be constant during light and integrating η (/D) i ln BSFC V i-1 i-1 Remarks: i -E BSFC V =ex 1000 η (/D) (3.39) (3.40) (i) As mentioned earlier, in a light with both (/D) and V as constant, the light altitude o airlane would increase as the weight o airlane decreases due to consumtion o uel. This is called cruise-climb. However, change in light altitude is small and light can be regarded as level light. (ii) To evaluate the uel raction or loiter o seciied duration, the values o BSFC, V, η and (/D) are required or airlanes with engine-roeller combination and those o TSFC and (/D) or the jet airlanes. (iii) Equation (3.37) shows that or a jet airlane the uel required or a seciied endurance, would be minimum when the light takes lace at C corresonding to (/D) max. Equation (3.40) shows that or an airlane with engine-roeller combination the uel required or given endurance would be minimum, when V/(/D) is minimum. Since, V is roortional to 1/( C (C D / C ) should be minimum or ( C 3/2 3/2 ), this imlies that /C D ) should be maximum, or uel required to be minimum. For a arabolic olar (C D = C Do + KC ), it can be shown that value o C corresonding to ( C corresonding to this value or C is (/D) max. 3/2 2 /C D ) max is (3C DO /K). The value o (/D) Det. o Aerosace Engg., Indian Institute o Technology, Madras 8

9 Chater Estimation o (/D) max outline o aroach Raymer (Reerence 1.18, chater 3) inds that (/D) max or a chosen tye o airlane deends on the wetted asect ratio (A wet ) deined as : 2 b A = (3.41) S wet wet where, b = wing san and S wet = wetted area o the airlane. Further S wet, or a chosen tye o airlane, is a multile o S the wing area. Using Figs.3.5 and 3.6 o Re.1.18 one can get a rough estimate o (/D) max. However, keeing in view the need or drag olar or otimization o wing loading 2 in chater 4, the exressions or drag olar (C D = C DO + KC ) are deduced which aear to be adequate at this stage o reliminary design. The general exressions or C DO and K obtained in Re.1.15 chater 6 are used or this urose. Though Re.1.15 gives general exression or C DO and K or both subsonic and suersonic airlanes, here the attention is ocussed on subsonic airlanes. Reerence 1.15 chater 6 gives the ollowing exressions or C DO and K or subsonic airlanes with wings o moderate to high asect ratio (A>5). C D0 = RwT S M C l M(cosΛ 1/4 ) Rw A-(t/c) t (λ)a(10 ) 1 6 c 0.1(3N e +1) K = { ( M ) 1+ + } πa (4+A) cos Λ 1 4 where, A = wing asect ratio M = Mach number S = wing area t/c = wing thickness ratio λ = taer ratio o wing 1/4= quarter chord swee o wing N e = number o engines, i any, located on to surace o wing (3.42) (3.43) Det. o Aerosace Engg., Indian Institute o Technology, Madras 9

10 Chater-3 A = airoil actor. A value o 0.93 is suggested or secial airoils and 0.75 or older (NACA) airoils. C l = unction o airoil chord over which the low in laminar. R w = S wet / S T = a actor which is unity or a very streamlined shae and takes into account the increase in C D due to dearture rom streamlined shae. = A actor which gives correction or wing thickness ratio and is given by : 3 R t/c R R 0.25 w = w w λ = actor which takes into account eect o wing taer ratio. (3.44) = [1+1.5( λ - 0.6) 2 ] (3.45) Once C D0 and K are obtained, (/D) max is given by : 1 /D = (3.46) 2 C K max D0 Three tyical cases viz. a high subsonic jet airlane, a turboro airlane and a low seed iston-engined airlane are considered. Based on Eqs.(3.42) and (3.43) drag olars are deduced or these three cases. Suggestions are also given to obtain drag olars o similar airlanes Drag olar o tyical high subsonic jet airlane A tyical high subsonic jet has the ollowing eatures: (a) M = 0.8, (b) A = 9, (c) an advanced suercritical airoil with t/c o 0.14, (d) taer ratio o 0.25, (e) swee 1/4 o 30 o, () N e = 0, (g) due to high Reynolds number the low can be treated as turbulent almost rom the leading edge or C l = 0. From chater 6 o Re.1.15 the wetted area o such airlanes is 5.5 times the wing lanorm area or R w = 5.5. Further, T is 1.1. Hence, in the resent case, with the above chosen arameters, the ollowing values are obtained. Det. o Aerosace Engg., Indian Institute o Technology, Madras 10

11 Chater-3 2C R l (I) 1- = 1.0. w (II) A - (t/c) = = (III) = = Remark: For this category o airlanes the eect o t/c on can be obtained by substituting dierent values o (t/c) in Eq.(3.44). The ollowing values or are obtained. t/c (IV) λ = [ ] = (V) 20 1) (0.866) M(cosΛ 20 = = A - t/c (VI) 20 McosΛ ( 1) M = = A -(t/c). Consequently, C = S D = S (3.47) The eect o swet angle on C D0 is o secondary nature. However, it is observed that C D0 should be reduced by about 0.4% or each increase in swee by 1 o. The eect o asect ratio on C DO is negligible. Thus, or o Λ 1 =30, A = 9 and t/c = C D0 = S (3.48) From Eq.(3.48) the variation o C D0 with S is as ollows. Det. o Aerosace Engg., Indian Institute o Technology, Madras 11

12 Chater-3 S(m 2 ) C D Remark: For Boeing 747 with S = 511 m 2, and 1/4 38.5, these arameters would give C DO o This value is close to the actual value or the airlane. See also Aendix C in Re. 3.1 Value o K : The terms in Eq.(3.43) are now evaluated or tyical high subsonic jet airlane with eatures mentioned under items (a) to (g) at the beginning o this subsection. The ollowing quantities are obtained. (I) M 6 = 1 = 0.12 (0.80) 6 = (II) (III) t λa(10 ) c (1.4) =1+ cos Λ cos o 1/4 0.1(3N e +1) 0.1(0 +1) = = (4 + A) 4+9 Hence, Eq.(3.43) yields: = 1+ = cos 30 1 K = = π 9 Exressing Remarks: 1 K= πae, gives e = (i)angle o swee does have signiicant eect on K. For M = 0.8, t/c = 0.14, =0.25, N e = 0 and C l = 0, Eq.(3.43) gives: K= = π A cos Λ1/4 πa cos Λ1/4 (3.49) Det. o Aerosace Engg., Indian Institute o Technology, Madras 12

13 Chater-3 For an airlane with S = 100 m 2, 1/4 30 o, Eqs.(3.48) and (3.49) give the ollowing drag olar. C D = C Hence, /D max = = = C K Do This value o (/D) max is tyical o such airlanes(re.1.18, Chater 3). i) Boeing 787-Dreamliner being brought out by Boeing has very smooth surace, o t/c = 11%, Λ =32, S = 331 m 2 and asect ratio o 10.4 with winglets at average 1/4 wing tis. It is estimated to have a C D0 o and K o 0.04 resulting in (/D) max o 22. It may be added that the eect o winglets on reducing induced drag can be estimated aroximately by adding hal the height o winglet to the wingsan (Re.1.15, chater 5, see also section o Re.3.3) Det. o Aerosace Engg., Indian Institute o Technology, Madras 13