Aircraft Design 1 (1998) 193}203

&&Starting mass''*a complex criterion of quality for aircraft on-board systems Yury M. Shustrov Moscow Aviation Institute, Volokolamskoe Shosse, 125871 Moscow, Russia

Abstract

It is very important to have minimum weight of aircraft equipment. It allows to increase the payload or #ight range or to improve some other aircraft characteristics. For the systems that are consuming energy during the #ight it is not less important to save the power spent on their functioning. &&Starting mass'' gives the opportunity to combine these two heterogeneous sides of the equipment properties into one complex characteristic, which depends on the #ight duration and on the aircraft engine parameters. The main idea in solving the problem of combining mass and energy in one complex parameter arises from the fact that practically the only on-board source of energy is the fuel stored in the aircraft tanks. Therefore, the energy consumed by the system may be substituted by the equivalent mass of fuel. Another component of the &&starting mass'' is the fuel spent on the transportation of the "rst two parts. The necessity of this third part arises from the di!erences in fuel expenditure on transportation of unchangeable and changeable loads during the #ight. The summation of these weight components is called &&starting mass'' because all of them should be present in the aircraft before take-o!. The core of the method was developed by Bulaevsky and myself. It is re"ned and partially corrected in this paper.  1999 Elsevier Science Ltd. All rights reserved.

1. Estimation of fuel 6ow spent on various forms of on-board systems energy consumption Fuel is almost the only source of aircraft on-board energy. The following obvious formula can be used for calculation of the fuel #ow m necessary to produce the power N:  N , m "  g H  S

(1)

where H is the calori"c value of the fuel, and g is the overall e$ciency of fuel energy transformaS  tion into power of a determined type. 1369-8869/98/$ } see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 1 36 9- 88 6 9 (9 8 )0 0 01 6 - 0

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1.1. Fuel yow, required for consumption of ram air from a special air intake or from the engine air intake (stagnation of ram yow) The simplicity of Eq. (1) hides behind g all the energy transformation process peculiarities.  Therefore sometimes (as shown below) it is easier to obtain special dependencies instead of Eq. (1). The scheme of an air-to-air heat exchanger is presented in Fig. 1. Every air intake causes drag force X that can be divided in two parts X"X #X , (2)   where X is the total aerodynamic drag, X the external inlet body drag, and X the ram drag,    < DF , (3) X "c o  V',  2 @ where c is the drag coe$cient, o the free air density, < the aircraft #ight velocity (approximV',   ately equal to < in the "gure), and F the inlet body maximum cross-sectional area. 
Usually, the drag component X is relatively small and in most cases may be neglected.  The drag component X can be calculated in accordance with the impulse conservation law:  X "m < !m < cos a , (4)  ',  ',  where m is the inlet air #ow, < the exit velocity of the air after passing through the system (at the ',  nozzle exit plane) and a the angle between the vectors < and < .   To compensate this drag, additional engine thrust and corresponding additional fuel #ow m ', is required: m "c X"c (m < !m < cos a) , ', 2 2 ',  ',  where c is the fuel consumption per unit thrust. 2 For piston or turboprop engines,

(5)

m "clX< "cl< (m < !m < cos a) ,   ',  ',  ',

(6)

Fig. 1. Ram air #ow through a heat exchanger (schematic).

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where cl is the fuel consumption per unit of engine power. Variant of Eqs. (5) and (6): m "bc m < "bcl m <  , ',  ', 2 ',  where b is a coe$cient calculated by < cos a b"1!  <  or assumed from statistics (with ranges 15b50).

(7)

(8)

1.2. Fuel yow, required for consumption of air compressed in a turbojet engine compressor The simplest scheme of a turbojet engine is shown in Fig. 2. The power needed for air compression can be determined with the expression N "c m D¹ , (9)  N   ¹ (nA\A!1) D¹ "    , (10)  g  where c is the speci"c heat of air at constant pressure, m the compressed air #ow; D¹ the N   compressor temperature di!erential; ¹ the compressor inlet temperature, n the pressure ratio    ("p /p ), c the ratio of speci"c heats, and g the compressor isentropic e$ciency.     (In formulae (9) and (10) as well as in corresponding downward formulae the total temperatures and pressures are used). Using Eq. (9) one can easily obtain from Eq. (1) the fuel #ow m necessary for air compression:  c m D¹ (11) m " N  .  H g   In this case g "e g g g , 
  

(12)

Fig. 2. Schematic diagram of a turbojet engine: c } compressor; bc } burning chamber, t } turbine; n } nozzle; ¹ }¹ } temperatures at given cross-sections.  

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where e is the burning completeness coe$cient, g the burning chamber heat losses coe$cient,
  g the thermal cycle e$ciency and g the mechanical e$ciency (mechanical losses from turbine 

to compressor), ¹ !¹ , (13) g "   ¹  ¹ !¹ #¹ !¹ D¹ #D¹ D¹ D¹   "  "  1#  g "  (14)  ¹ ¹ ¹ D¹     where D¹ is the turbine temperature di!erential and D¹ the engine outlet nozzle temperature   di!erential. Substitution of Eq. (12) into Eq. (11) yields






c m D¹ c m D¹ N   ¹ . (15) m " N   "  H e g g g H e g g D¹ (1#D¹ /D¹ )  S
     S
   The correlation between turbine temperature di!erential D¹ and compressor temperature  di!erential D¹ may be revealed from the power balance condition:  N "g N , (16) 

 N "c m D¹ , (17)  N  N "c m D¹ , (18)  N    c m D¹ " N   g D¹ , (19)  c m  N where m is the air #ow through the compressor, c the combustion chamber exhaust #ow speci"c N  heat at constant pressure, m the gas #ow through the turbine; m "m !m #m , m the bleed air  

#ow (e.g. for air conditioning system) and m the fuel #ow to the combustion chamber.  Now Eq. (15) may be expressed as c m¹ m c m ¹ N    + N    . (20) m "  H e g (1#D¹ /D¹ ) m H e g (1#D¹ /D¹ ) S
 O   S
 O   In case of mid-compressor air bleed, D¹ in Eq. (9) should be superseded with D¹ , equal to 
1 D¹ "¹ (nA\A!1) , (21)
 
g
while 1 (22) D¹ "¹ (nA\A!1) ,     g  thus nA\A!1 , (23) D¹ "D¹

 nA\A!1 

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and "nally nA\A!1 c m¹ N   
, (24) m " 
H e g (1#D¹ /D¹ ) nA\A!1 S
 O    assuming that the compressor e$ciency is the same for the compressor stages (g "g ).
 The result can be easily extrapolated to di!erent engines: turboprop, bypass, multishaft and to multistage turbines. Two main schemes and their combination may be considered. 1. An additional load (propeller, fan or bypass compressor) is driven with a special shaft from separate stage(s) of the turbine. Then additional temperature di!erentials in the thermodynamic cycle should be taken into account:






D¹ D¹ G#  . (25) D¹R"D¹ # D¹ #D¹ "D¹ 1#  G   D¹ D¹   G G 2. A shaft power o!take N for an additional load is supplied in a known proportion by the  same turbine (see Eq. (16)): g N "N #N .

   Then one can get in a manner similar to Eqs. (16)}(18)

(26)

g D¹ c m

 , D¹ " N    c m (1#N /N )   N and "nally for the combination of the cases mentioned above:

(27)

nA\A!1 c m¹ N    ;
. (28) m "  D¹ N D¹ nA\A!1 G    1# # H e g 1# S
 O D¹ N D¹    G In formula (28) only the engine compression energy is taken into account. The full energy (and equivalent fuel #ow) must include the part connected with the corresponding ram air consumption:









m "m #m ,     nA\A!1 c m¹ N    ;
. m "bc m < #  2   A\A!1 N D¹ D¹ n G  H e g 1#  1# # S
 O N D¹ D¹    G









(29)

1.3. Fuel yow, required for engine shaft owtakes In this case, formula (15) should be slightly changed: N  . m " , H e g g g S
 / 

(30)

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In a manner similar to the one used previously one can get N  m " ¹ , (31) ,  D¹ D¹ G H e g g D¹ 1# # S
 O  D¹ D¹   G N  , (32) m " , D¹ D¹ G  # H e g g g (1!n\A\A) 1# S
 O   D¹ D¹   G ¹ gN  .   m " (33) , D¹ ¹ D¹ G   H e g (nA\A!1) 1# # S
 O  D¹ D¹   G When the necessary information to use formulae (31)}(33) is not known, it is possible at a "rst approximation to use the simplest equation (1) with g "0.2 }0.3 (which is typical to many modern  heat engines) or to use the equivalent relation
















m "cl N , ,  with cl (fuel consumption per unit of engine power) taken in the range 0.27}0.4 kg/kW/h.

(34)

2. Estimation of fuel quantity, spent on transportation of loads with constant or decreasing mass during the 6ight After estimation of the fuel #ow necessary for the system energy consumption, one can calculate the fuel quantity M demanded for the whole #ight: , O (35) M " m & dq"m q ,   ,   where q is the time, q the #ight duration, m & the summarized instantaneous fuel #ow for all types   of energy consumption and m the mean fuel #ow during the #ight for systems energy consump tion (it will be assumed hereafter that m "m ).   Two components of the system mass are now distinguished: 1 } system dead mass and 2 } fuel mass M to be spent on system functioning. ,



2.1. Calculation of the fuel quantity for transportation of mass M which is constant  during motion At an arbitrary moment during the #ight the system has the mass M"M #M (36)   where M is the fuel mass required for the transportation of M during the #ight time left.   Time derivative of Eq. (36) looks like dM dM , " dq dq

(37)

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in turn ¹ dM M "!c ¹ "!c M  "!c M¹M 2  2 2 M dq M ! !

(38)

or N dM M "!cl N "!cl M  "!cl MNM . (39)  M M dq ! ! Taking into account Eqs. (38) and (39), di!erential equations can be obtained from Eq. (37), dM"!c M¹M dq 2 dM"!cl MNM dq

(40) (41)

Eqs. (40) and (41) can be integrated over time from 0 to q :  M M "M [eA22 O!1]   or

(42)

(43) M "M [eAl,M  O!1] .   In Eqs. (38)}(43), M is the fuel quantity for M transportation during total time of #ight, kg, q is    the #ight (or considered part of #ight) duration, h, ¹M "gC /C is the current thrust loading, N/kg, " * NM "g< C /C the current power loading, kW/kg, g the acceleration due to gravity, C the current  " * " drag coe$cient, C the current lift coe$cient, M the aircraft mass at considered moment, kg, * ! N and ¹ are the current power or thrust of power plant, kW or N, (c ¹M ) and (clNM ) are the

  2 mean values of the products for #ight duration, and as it was previously mentioned, c is the fuel 2 consumption per unit thrust, kg/N/h and cl the fuel consumption per unit of engine power, kg/kW/h. For estimation of the exponents in Eqs. (42) and (43) it is appropriate to note that they were obtained from integrals of the following type: O c ¹ 2  dq . (44) M ! M "M /M (In fact, it is The result of this integration is very similar to the cruise fuel fraction M    slightly higher due to the aircrafts mass reduction during the #ight.) This indicates that to a good approximation the exponents (c ¹M ) q and (clNM ) q are equal to the aircraft cruise mass factor, 2 

 which is nearly always between 0 and 0.5. It is also worth noting that the exponents (c ¹M ) and 2 (clNM ) are completely equivalent and in case of necessity may rightfully substitute one another.



2.2. Calculation of fuel quantity for transportation of mass which is decreasing during the yight De"ne a cooling agent with initial mass equal to M M m " .  q 



, that is spent evenly during the #ight: (45)

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After proper transformations and integration the following is obtained: M   [eA22M  O!1]!M M "   (c ¹M ) q 2 

(46)

M  [eA22M  O!1]!m ) q . M "    (c ¹M ) 2

(47)

or

2.3. Calculation of total fuel quantity M spent on all types of energy consumption summarized , with corresponding transportation fuel Corresponding to Eq. (35), the fuel quantity covering system energy consumption is equal to M "m q . The assumption of constant fuel #ow made in Eq. (35), base on the theorem of mean ,   values, allows to employ Eqs. (46) and (47) for the considered problem with substitution of m by  m : , m (48) M "  [eA22M  O!1] . , (c ¹M ) 2 3. Conclusion After de"ning all components one can "nd the total mass of the system: (49) MR"M #M #M #M #M .     , The system has this mass (including the components that will be spent) before the vehicle starts its motion. Therefore, it has ben called starting mass. After insertion of the corresponding expressions into each term on the right-hand side of Eq. (49), the "nal equation for the starting mass gets the form eA22M  O!1 MR"M eA22M  O#(M #M )   , (c ¹M ) q 2 

(50)

or m #m  [eA22M  O!1] . MR"M eA22M  O#  (51)  (c ¹M ) 2 The starting mass may be considered as a self-su$cient criterion of quality in cases when aircraft mass saving is most important. It does not contradict other quality characteristics and can also be used as a part of more complicated criteria, e.g. economical. On the basis of the starting mass some other criteria can be obtained. For example, if the considered system may hypothetically be taken away from the aircraft and substituted by some constant payload (not consuming energy) its mass can be considered equivalent to the system.

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The condition of equivalence is M #M "MR , (52)   where MR is the starting mass of the system, M the equivalent mass of the system (imaginary  payload) and M the fuel quantity for transporting equivalent mass.  The fuel quantity M for transporting equivalent mass can be calculated by Eq. (42):  M M #M [eA22 O!1]"MR , (53)   whence M "MR e\A22M  O .  Combination of Eq. (54) with (51) yields M "M #(M #M )    

(54)

1!e\A22M  O (c ¹M ) q 2 

(55)

or m #m  [1!e\A22M  O] M"M #    (c ¹M ) 2

.

(56)

For in"nite #ight time the equivalent mass from Eq. (56) approaches the sum m #m  . (57) M "M #    (c ¹M ) 2 This allows to compare two variants of the system at "rst approach using an approximate relation which says: the second variant is better than the "rst if (m !m )#(m !m )    . (58) M !M '     (c ¹M ) 2 It is possible to derive other criteria similar to the starting mass, if one takes into account that in addition to the system and its associated fuel it is necessary to add corresponding masses of fuselage, lifting surfaces, undercarriage, etc., and on transportation of all these parts appropriate fuel is spent. Realization of such an approach allows to determine a part DM  of the aircraft  take-o+ mass associated with the considered system 1#A DM"MR ,  1!A [eA22M  O!1]

(59)

where A is the ratio of the aircraft mass required to accommodate and transport a given amount of useful load (payload and fuel) to the mass of that useful load itself. The choice of any of the above-mentioned system quality criteria can be de"ned by reasons of convenience of calculations, presentation of results or any other additional motives.

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Table 1 Compared drives characteristics System

Weight (kg)

Input power (kW)

Fuel consumption for system functioning (kg/h)

Electric Hydraulic Pneumatic

92.5 78.3 44.5

43.6 50.4 1000 kg/h of bleed air

23.6 26.6 38.7

Fig. 3. Results of calculating starting mass for drives of di!erent type: E } electric; H } hydraulic; P } pneumatic.

In summary, it is important to note that the considered parameters, as they are visible in the presented formulas, depend on the characteristics of the aircraft and engine. Therefore, a situation is at least theoretically probable, where two comparable variants can exchange their order of favour when di!erent aircrafts are compared. The analysis of given dependencies shows, in particular, that for aircraft with greater power (thrust) loading (for example, helicopters or high-speed planes in comparison with subsonic planes) the economy of weights is the rather more important problem, than economy of energy consumption.

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3.1. Example Comparison of hydraulic, electric and pneumatic drives (Table 1) with the same output power of 30 kW and the same length of communications equal to 15 m. 3.2. Assumed aircraft characteristics Flight Mach number !0.8, #ight altitude !11,000 m, thrust loading !1 N/kg, fuel consumption per unit thrust of turbojet engine !0.1 kg/N/h, compressor pressure ratio !8, compressor e$ciency !0.82, combustion chamber exhaust temperature !1300 K, burning completeness coe$cient !0.98, burning chamber heat losses coe$cient !0.95, mechanical e$ciency (mechanical losses from turbine to compressor) !0.98, calori"c value for fuel !44 MJ/kg. Calculations were made with the use of the above formulae. The results are presented by the graphs in Fig. 3. It can be seen that the starting mass of the systems changes with the increase of #ight duration. For short #ight duration the use of pneumatic drive is more pro"table as this system has the least starting mass. Then for the #ight duration of approximately 3.3 h the hydraulic drive becomes preferable. And at last, for #ight duration more than 5 h electric drive proves to be the best (with small di!erence from hydraulic drive). The graphs show that for #ight duration of 10 h the starting mass is 7}17 times more than dead mass of the considered systems.

References 1. Shustrov Y, Bulaevsky M. Aircraft airconditioning systems. Moscow: &&Mashinostroenie'', 1978: 160 pp.