3 Mission Analysis
3.1 Concept
With preliminary design values of takeoff thrust loading (TsL/Wro) and wing loading (Wro/S) now in hand, the next step is to establish the scale of the aircraft via the estimation of gross takeoff weight (Wro). This will be accomplished by flying the aircraft through its entire mission on paper.
The key fact is that WTo is simply the sum of the payload weight (We), the empty weight (WE), and the required fuel weight (WF), or
WTO = W p -t- WE -t- WF (3.1)
These will now be considered in turn.
Wp is specified in the Request for Proposal (RFP) and comes in two parts. The first is the expendable payload weight, which is delivered during the trip (WpE), such as cargo or ammunition. The second is the permanent payload weight, which is carried the entire mission (Wpp), such as the crew and passengers and their personal equipment.
WE consists of the basic aircraft structure plus any equipment that is permanently attached, such as the engines, the avionics, the wheels, and the seats. In short, WE includes everything except Wp and WF. WE can be estimated as a fraction of Wro, as shown in Figs. 3.1 and 3.2, which correspond to conventional, lightweight metal construction. The ratio of WE to Wro has obviously depended on the type of aircraft and its size, but the range of this parameter is not wide. Because WE/Wro varies slowly with Wro, an initial value may be obtained from an initial estimate of WTO and any necessary correction made when WTO becomes more accurately known.
When one contemplates the enormous range of ages and types of aircraft dis- played in Figs. 3.1 and 3.2, and the relatively narrow range of WE/WTo, it is possible to conclude that "practical" aircraft have "natural" empty weight frac- tions that provide reliable future projections. This observation should increase your confidence in their validity.
WF represents the fuel gradually consumed during the entire mission. Except for the instantaneous release of WpE, the aircraft weight decreases at exactly the same rate at which fuel is burned in the engine. The rate of fuel consumption, in turn, is simply the product of installed engine thrust (T) and installed engine thrust specific fuel consumption (TSFC). T can be found from whatever version of Eq. (2.1) is most convenient, while TSFC depends on the engine cycle, flight conditions, and throttle setting and must initially be estimated on the basis of experience.
The fuel consumption analysis has several fortunate benefits. For one thing, it results in calculations based on relatively little information. For another, it reveals
55
56 AIRCRAFT ENGINE DESIGN
0.70 0.65 0.60 0.55
0.50
0.45 0.40 0.35 0.30 0
Fig. 3.1
i i i
C-7A 1D
£ -12A C-123
• ~ • 7 6 7 - 2 0 0
~ o • C-9A
777-200
A310 •
UV-18 •
C-21A A300 A330
0 C - 1 4 0 • •
757-200 • L-1011
S-3 O 777-300ER
767-400ER A340
',-20A • C-130 • C-5B 7 4 7 - 4 0 0
• C-170 C.5A O• ~
Concorde
747-300 C-141B • • KC-10A
B-1B • KC-135R •
A380
KC-135A •
, I , I , I , I , I , I ,
200 400 600 800 1000 1200 W (1,000 lbfs)
TO
Weight fractions of cargo and passenger aircraft, l'z 1400
the "best" way to fly certain legs for minimum fuel usage. Finally, it shows that the fuel burned during each mission leg is a fraction of the aircraft weight starting the leg, whence WF becomes a calculable fraction of Wro. Most of the analysis below is devoted to the development of the "weight fraction" equations needed for the many possible kinds of mission legs.
Substitution of the derived results for We/Wro and Wr/Wro into Eq. (3.1) yields an unambiguous value for Wro as a function of Wp,o and Wee (see
0.80 0.75 0.70 0.65
°.6°
0.55 0.50 0.45
o4ol
10 Fig. 3.2
I I
• X-29A
F-102A F- 16C/D
F-16A/B
F-104C YF16
O • YF17
F-15
F-106A • F-15C
F-105G F-14A F-117A •
F-100D • F-4E •
F-IO1B
F/A-18E
• F-14B
I
20
I I I F I I
30 40 50 60 70 80 W (1,000 lbfs)
TO
Weight fractions of fighter aircraft. 12
F-11
I
90 100
MISSION ANALYSIS 57 Sec. 3.2.12). Then TSL and S are found by multiplying W r o by thrust loading and wing loading, respectively.
dW d t which may be rewritten as 3.2 Design Tools
The following material might very well be called the "thermodynamics of flight"
because it deals largely with the way the thrust work of the engine is used by the aircraft. We expect that it should have a special appeal to engine designers, who always deal with the conversion of energy from one form to another, but seldom see aircraft treated in this manner.
We begin by considering the rate at which aircraft weight is diminishing as a result of the consumption of fuel, namely
d W F
. . . . T S F C × T (3.2)
dt
dW T
- - - - T S F C - - d t (3.3)
W W
Note again that T is the installed thrust and T S F C is the installed thrust specific fuel consumption. Also, since
T T dt T d s
dt -- - - ds --
W W d s W V
then this portion of Eq. (3.3) represents the incremental weight-velocity specific engine thrust work done as the amount of fuel dWF is consumed. Just as it is with any other thermodynamic situation, this engine thrust work will be partly invested in the mechanical energy (potential plus kinetic) of the airplane mass and partly "dissipated" into the nonmechanical energy of the airplane/atmosphere system (e.g., wing tip vortices, turbulence, and aerodynamic heating). The ratio of mechanical energy to "dissipation" will, of course, depend on the type of flight under consideration, as will be seen in what follows.
The main work of this section will be to determine how Eq. (3.3) can be integrated in order to obtain "weight fractions"
Wf a/ __. Wl W i , itial Wi
for a variety of mission legs of practical interest. From a mathematical point of view, proper integration of Eq. (3.3) requires a knowledge of the behavior of the thrust specific fuel consumption and the "instantaneous thrust loading"
{ T / W = ( o t / ~ ) ( T s L / W r o ) } as a function of time along the flight path. Experience indicates that the integration can be separated into two distinct classes, cor- responding to Ps > 0 (type A) and Ps = 0 (type B). These will now he considered in turn.
Instantaneous thrust loading behavior: type A, Ps > O. When Ps > 0, some of the thrust work is invested in mechanical energy. Also, it is generally true that specific information is given regarding the amount of installed thrust applied,
58 AIRCRAFT ENGINE DESIGN
as well as the total changes in altitude (h) and velocity (V) that take place, but not the time or distance involved. In fact, the usual specification for thrust is the maximum available for the flight condition, or T = a TsL. Examples of type A are found as Cases 1-4: 1) constant speed climb, 2) horizontal acceleration, 3) climb and acceleration, and 4) takeoff acceleration.
Progress toward a solution may now be made by using Eq. (2.2a) in the form 7" V dt = 7 " ds = d(h + V2/2go) _ dze (3.4)
W W 1 - u 1 - u
where
D + R
u -- - - (3.5)
T whence, combining Eqs. (3.3) and (3.4),
d W TSFC
- - -- - - d(h + V2/2go) (3.6a)
W V(1 - u )
o r
d W TSFC
- - -- - - dze (3.6b)
W V(1 - u )
The quantity u determines how the total engine thrust work is distributed between mechanical energy and dissipation. More precisely, Eq. (3.5) shows that u is the fraction of the engine thrust work that is dissipated, so that (1 - u) must be the fraction of the engine thrust work invested in mechanical energy. This is further confirmed by Eq. (3.4), which reveals that (1 - u) equals the change in mechan- ical energy (W dze) divided by the total engine thrust work Tds. Note that when T = D + R and u = 1, all of the thrust work is dissipated, and the type A analysis yields no useful results.
The actual integration of Eq. (3.6a) is straightforward and depends only upon the variation of {TSFC/V(1 - u)} with altitude and velocity. When this quantity remains relatively constant over the flight leg, as it frequently does, the result is
~-/Wf- exp { -- v ~ T S F C ( V 2 ) } u ) ~go ~ A h + (3.7a)
o r
- - _ I TSFC |
Wfwi -- exp [ - V(1 - u) AZe I (3.7b) where AZe is the total change in energy height. Otherwise, the integration can be accomplished by breaking the leg into several sm~ller intervals and applyng Eq. (3.7) to each. The overall W f / W i will then be the product of the results for the separate intervals.
Equations (3.6) and (3.7) highlight the fact that, within certain limits, potential energy and kinetic energy can be interchanged or "traded." For example, if there were no drag (u = 0) and Ze were constant, as in an unpowered dive or zoom climb, no fuel would be consumed and dW = 0 or W f = Wi. This, in turn, reveals the
MISSION ANALYSIS 59 forbidden solutions of Eqs. (3.2), (3.6), and (3.7) for which W would increase during the flight leg and which correspond to T < 0 and dZe < 0. Lacking special devices onboard the aircraft that could convert aircraft potential and/or kinetic energy into "fuel," it must be true for any condition of flight that dW < 0 and w: <_wi.
fs a n d t h e m i n i m u m fuel path. As already mentioned, the fuel consumption analysis also reveals the "best" way to fly type A legs for minimum fuel usage.
Equations (3.2), (2.2b), and (2.3) may be combined to yield ot TSFC dWF = T x TSFC x dt = TsL dZe
Ps where
T - ( D + R ) } T P ~ = ~ V = ~ ( 1 - u ) V Defining the fuel consumed specific work (fs) as
Ps TSL dze
fs "-- - - -- (3.8)
ot TSFC dWF
then
f2 f Ze2 dz e
WF~ 2 = dWF = TsL (3.9)
tl Z e l Is
Equation (3.9) shows that the minimum fuel-to-climb flight from
Zel
(hl, Wl) to Zea (he, V2) corresponds to a flight path, that produces the maximum thrust work per unit weight of fuel consumed at each energy height level in the climb, i.e., the maximum value of fs at each Ze. As you can see from Eq. (3.8), the units of fs are feet.Constant fs and Ze contours in the altitude-velocity plane (e.g., see Fig. 3.E6 in Sec. 3.4.4) provide a graphical method for finding the maximum f~ at any Ze and hence the minimum fuel-to-climb path from Zel to Ze2. This is analogous to using the P~ and ze contours of Fig. 2.E5 to find the minimum time-to-climb path in conjunction with the time-to-climb equation from Eq. (2.2b).