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G(W,V, h)dW (4.22) where W1 = Wf , Wn+1 = W0. (4.23) Then, the distance factor and the time factor are assumed to vary linearly with the weight over each subinterval, that is, F = Fk + Fk+1−Fk Wk+1−Wk (W −Wk) G = Gk + Gk+1−Gk Wk+1−Wk (W −Wk) (4.24) so that Eqs. (4.22) can be integrated analytically to obtain xf − x0 = n∑ k=1 1 2 (Fk+1 + Fk)(Wk+1 −Wk) (4.25) tf − t0 = n∑ k=1 1 2 (Gk+1 +Gk)(Wk+1 −Wk). (4.26) 88 Chapter 4. Cruise and Climb of an Arbitrary Airplane In general, the number of intervals which must be used to get a reason- ably accurate solution is small, sometimes just one. While the distance and time can be computed for different velocity profiles such as constant velocity or constant lift coefficient, it is important for design purposes to find the maximum distance trajectory and the maximum time trajectory. 4.8 Cruise Point Performance for the SBJ The analysis of the distance factor F and the time factorG is called point performance because only points of a trajectory are considered. For a fixed altitude, this can be done by plotting F and G versus velocity for several values of the weight. Regardless of the weight interval [W0,Wf ] that is being used to compute distance and time, F and G can be com- puted for all values of W at which the airplane might operate. The use of F and G to compute the distance and time for a given velocity profile V (W ) and a given weight interval [W0,Wf ] is called path performance, because a whole path is being investigated. Point performance for the SBJ begins with the solution of Eq. (4.14) for the power setting P (h, V,W ) using Newton’s method. It is shown in Fig. 4.3 for h = 35.000 ft. Note that the power setting is around 0.90. Then, P is substituted into Eqs. (4.17) to get the distance factor F (W,V, h) and the time factor G(W,V, h). Values of F and G have been computed for many values of the velocity (1 ft/s intervals) and for several values of the weight. These quantities are plotted in Figs. 4.4 and 4.5. It is observed from Fig. 4.4 that the distance factor has a maximum with respect to the velocity for each value of the weight. This maximum has been found from the data used to compute F . At each weight, the velocity that gives the highest value of F is assumed to rep- resent the maximum. Values of V (W ), Fmax(W ), and the corresponding values of G(W ) are listed in Table 4.1 and plotted in Figs. 4.6 and 4.7. Note that V and Fmax(W ) are nearly linear in W . It is observed from Fig. 4.5 that the time factor has a maximum with respect to the velocity for each value of the weight. This maximum has been found from the data used to compute G. At each weight, the velocity that gives the highest value of GF is assumed to represent the 4.8. Cruise Point Performance for the SBJ 89 P 350 450 550 650 750 850 V (ft/s) W = 9,000 lb W = 11,000 lb W = 13,000 lb h = 35,000 ft .84 .88 .92 .96 1.00 Figure 4.3: Power Setting (SBJ) 350 450 550 650 750 850 V (ft/s) F (mi/lb) W = 9,000 lb W = 11,000 lb W = 13,000 lb Maximum Distance Factor h = 35,000 ft .1 .3 .4 .5 .2 Figure 4.4: Distance Factor (SBJ) 90 Chapter 4. Cruise and Climb of an Arbitrary Airplane 350 450 550 650 750 850 V (ft/s) W = 9,000 lb W = 11,000 lb W = 13,000 lb Maximum Time Factor h = 35,000 ft G x 103 (lb/hr) 0.6 0.8 1.0 1.2 1.4 Figure 4.5: Time Factor (SBJ) maximum. Values of V (W ), Gmax(W ), and the corresponding values of F (W ) are listed in Table 4.1 and plotted in Figs. 4.6 and 4.7. Note that V and Gmax(W ) are nearly linear in W . Note that the velocity for maximum distance factor is roughly 35% higher than the velocity for maximum time factor. 4.9 Optimal Cruise Trajectories At this point, there are two approaches which can be followed. One is to specify a velocity profile, say for example V= Const, and compute the distance and time for a given W0 and Wf . The other is to find the velocity profile V (W ) that optimizes the distance or that optimizes the time. Because distance factor has a maximum, the optimal distance trajectory is a maximum. Similarly, because the time factor has a max- imum, the optimal time trajectory is a maximum.. Optimal trajectories are considered first because they provide a yardstick with which other trajectories can be measured. 4.9. Optimal Cruise Trajectories 91 Table 4.1 SBJ Optimal Cruise Point Performance h = 35,000 ft Maximum Distance Maximum Time W V Fmax G V Gmax F lb ft/s mi/lb hr/lb ft/s hr/lb mi/lb 9,000 604 .442 1.073E-3 416 1.386E-3 .374 9,500 602 .434 1.058E-3 427 1.262E-3 .367 10,000 605 .424 1.031E-3 439 1.210E-3 .362 10,500 618 .415 .985E-3 456 1.161E-3 .361 11,000 631 .406 .943E-3 466 1.110E-3 .353 11,500 645 .398 .905E-3 477 1.064E-3 .345 12,000 658 .390 .870E-3 487 1.021E-3 .338 12,500 671 .383 .837E-3 496 .981E-3 .332 13,000 683 .376 .807E-3 506 .944E-3 .326 0.9 1.0 1.1 1.2 1.3 Wx10-4 (lb) V (ft/s) h = 35,000 ft. 350 450 550 650 750 Fmax Gmax Figure 4.6: Optimal Velocity Profiles (SBJ) 4.9.1 Maximum distance cruise The velocity profile V (W ) for maximizing the distance (4.20) is obtained by maximizing the distance factor with respect to the velocity for each 92 Chapter 4. Cruise and Climb of an Arbitrary Airplane 0.9 1.0 1.1 1.2 1.3 Wx10-4 (lb) Fmax (mi/lb) 300 x Gmax (hr/lb) .25 .30 .35 .40 .45 Figure 4.7: Maximum Distance and Time Factors (SBJ) value of the weight (see Sec. 4.3). This velocity profile is then used to compute the maximum distance factor which is used to compute the maximum distance and to compute the time factor which is used to compute the time along the maximum distance trajectory. This process has been carried out for the SBJ at h=35,000 ft with W0 = 12,000 lb and Wf = 10,000 lb (W0 −Wf = 2,000 lb of fuel). The values of Fmax and G shown in Table 4.1 are used to compute the maximum distance and the corresponding time from Eqs. (4.25) and (4.26). As an example, to compute the maximum distance and the corresponding time for h = 35,000 ft, W0 = 12,000 lb, and Wf =10,000 lb (2000 lb of fuel) using 5,000 lb weight intervals (n = 4), the values to be used in Eqs. (4.25) and (4.26) are listed in Table 4.2. Then, Eq. (4.25) gives the maximum distance of 813 mi (see Table 4.3). Similarly, the time along the maximum distance trajectory is obtained from Eq. (4.26) as 1.90 hr. This computation has also been made for one interval (n = 1), and the results agree well with those of n = 4. This happens because Fmax (Fig. 4.5) and G are nearly linear in W . 4.9. Optimal Cruise Trajectories 93 Table 4.2 Maximum Distance Cruise h = 35,000 ft, W0 = 12,000 lb, Wf = 10,000 lb k Wk Fmax,k Gk (lb) (mi/lb) (hr/lb) 1 10,000 .424 .001031 2 10,500 .415 .000985 3 11,000 .406 .000943 4 11,500 .398 .000905 5 12,000 .390 .000870 Table 4.3: SBJ Optimal Cruise Path Performance h = 35,000 ft, W0 = 12,000 lb, Wf = 10,000 lb 4 Intervals 1 Interval Maximum Distance (mi) 813 814 Distance Time (hr) 1.90 1.90 Maximum Distance (mi) 704 700 Time Time (hr) 2.20 2.23 To actually fly the maximum distance velocity profile, it is nec- essary to know the weight as a function of time. If it is not available, the optimal path can only be approximated. Other velocity profiles are possible: constant lift coefficient, constant velocity, constant power set- ting, etc. The importance of