The project refines the weight estimate and optimizes the design of an ultra-long-range business jet, using a sizing matrix and carpet plot.

1. Thomas Templin; MEM 425-001; September 2, 2016
Design Optimization & Carpet Plot
Refined Weight Estimate
Our team initially surmised that the takeoff gross weight of the aircraft we designed would be
around 100,000 lb, based on the published weight of the Gulfstream G650 ER, an aircraft with
somewhat similar requirements and specifications (Gulfstream G650ER, 2016, p. 3). Our initial
weight estimate, based on mission-segment fuel ratios, resulted in a takeoff-weight estimate of
99,821 lb (Raymer, 2012, pp. 27–47; Anderson, 1999, pp. 398–406). In order to arrive at a more
reliable estimate, our team decided to re-estimate the aircraft’s weight, in an iterative fashion,
after the configuration layout had been completed, using statistical methods based on regression
analysis and physics-based models, until convergence was achieved (Raymer, 2012, pp. 583–
594). The following equations for general-aviation weights were employed to compute
component weights:
𝑊wing = 0.036𝑆 𝑊
0.758
𝑊𝑓𝑤
0.0035
(
𝐴
cos2Λ
)
0.6
𝑞0.006
𝜆0.04
(
100 𝑡 𝑐⁄
cos Λ
)
−0.3
(1.5 𝑁 𝑊dg)
0.49
𝑊horizontal tail = 0.016(1.5 𝑁 𝑊dg)
0.414
𝑞0.168
𝑆ht
0.896
(
100 𝑡 𝑐⁄
cos Λ
)
−0.12
(
𝐴
cos2Λht
)
0.043
𝜆ht
−0.02
𝑊vertical tail = 0.073 (1 + 0.2
𝐻𝑡
𝐻𝑣
) (1.5 𝑁 𝑊dg)
0.376
𝑞0.122
𝑆vt
0.873
× (
100 𝑡 𝑐⁄
cos Λvt
)
−0.49
(
𝐴
cos2Λvt
)
0.357
𝜆vt
0.039
𝑊fuselage = 0.052𝑆𝑓
1.086
(1.5 𝑁 𝑊dg)
0.177
𝐿 𝑡
−0.051(𝐿 𝐷⁄ )−0.072
𝑞0.241
𝑊landing gear = 0.057𝑊dg
𝑊installed engine (total) = 2.575𝑊en
0.922
𝑁en
𝑊all else empty = 0.1𝑊dg
The following symbols were used in the above weights equations:

2. 2
A aspect ratio
Ht horizontal tail height above fuselage, ft
Hv vertical tail height above fuselage, ft
L/D lift-to-drag ratio
Lt tail length; wing quarter-MAC to tail quarter-MAC, ft
N limit load factor
Nen number of engines (total for aircraft)
q dynamic pressure at cruise, lb/ft2
Sf fuselage wetted area, ft2
Sht horizontal tail area, ft2
Svt vertical tail area, ft2
Sw trapezoidal wing area, ft2
t/c thickness-to-chord ratio
Wdg flight design gross weight, lb
Wen engine weight, each, lb
Wfw weight of fuel in wing, lb
Λ wing sweep at 25% MAC (subscript “t” or “h” for horizontal tail, “v” for vertical tail)
λ taper ratio (wing or tail)
During this process the fuel weight was fixed at 40,000 lb, in order to satisfy the range
requirement of 10,000 mi. This analysis produced a refined estimate of the takeoff gross weight
of 76,068 lb. The wing loading and thrust-to-weight ratio were also updated with each iteration,
and their final values came out to be W/S = 55 lb/ft2
and T/W = 0.22.
Design Optimization
In an attempt to potentially further reduce the aircraft’s weight (which was considered the
measure of merit), classical optimization using a sizing matrix and a carpet plot was performed
(Raymer, 2012, pp. 734–741). During this process, both the wing loading and the thrust-to-
weight ratio were systematically altered around their computed baseline values, and the aircraft
weight was re-estimated using the above-mentioned iterative algorithm. More specifically, the
following values were employed for these two performance parameters: W/S = 45, 55, or 65
lb/ft2
and T/W = 0.14, 0.22, or 0.26.
Whereas the new weight estimate was obtained by directly changing the W/S value in the
iterative loop of the MATLAB program, and thus not subjecting it to recalculation, increasing or
decreasing T/W from the baseline value involved selecting new engines, and thus more thrust and
a higher engine weight or less thrust at a reduced weight (Raymer, 2012, p. 117). For lowering
thrust (and thus T/W), the engine we originally chose (GE CF34-8C5) was replaced with the
model GE CF34-3 (SFC of 0.69 lb/lb/h), which weighs 1670 lb and provides 9220 lb of thrust at

3. 3
sea level (CF34-3 turbofan engine, 2016, p. 2). Two such engines would fulfill our previously
computed maximum-thrust requirement, while reducing performance and the safety margin in
the case of engine malfunction during takeoff. On the other hand, the model GE CF34-10A (SFC
of 0.65 lb/lb/h) was chosen as the heavier, more powerful engine (CF34-10A, 2016 p. 2). It
weighs 3700 lb and provides 17,640 lb of thrust. These engines, including our original choice,
power large business jet-type aircraft. The specifications of these two alternative engines were
used to calculate the new thrust-to-weight ratios (0.14 and 0.26, respectively) that were used in
the carpet-plot optimization method.
In addition to the takeoff weight, the takeoff distance, the landing distance, and the rate of climb
were recomputed upon completion of the iteration procedure, to verify their compliance with
design requirements and constraints. This process produced the sizing matrix shown in figure 1.
W/S
45 55 65
0.26 81,287 79,215 77,758 TOGW (lb)
2,752 2,806 2,857 sTO (ft)
1,653 1,761 1,869 sL (ft)
1,493 1,492 1,494 R/C (ft/min)
T/W 0.22 78,112 76,040 74,584 TOGW (lb)
3,128 3,182 3,233 sTO (ft)
1,636 1,740 1,844 sL (ft)
1,075 1,060 1,049 R/C (ft/min)
0.14 76,069 73,997 72,540 TOGW (lb)
4,514 4,568 4,619 sTO (ft)
1,624 1,725 1,826 sL (ft)
292 251 213 R/C (ft/min)
Fig. 1. Sizing matrix.
The entries of the sizing matrix show that the requirement for a landing distance of 3000 ft is
always met (third row for each T/W), and thus this constraint was not further investigated in the
carpet-plot optimization process. The reason for the relative insensitivity of the landing distance
to variations in W/S or T/W is presumably the only slight dependence of landing distance on W/S.
Moreover, thrust reversal was not employed in the landing-distance calculations.

4. 4
However, the sizing matrix also shows that the configuration using the lowest thrust-to-weight
ratio (T/W = 0.14) failed the requirements for takeoff distance and rate of climb (second and
forth rows). The reason for this observation is clearly attributable to the substantially lower thrust
of the CF34-3 engine, which fails to meet the acceleration and climb requirements laid out for
this project. It should be noted that our team used 1000 ft/min as the benchmark for the rate-of-
climb requirement, which was given as an alternative option in the document containing the
requirements/specifications for the project.
In order to provide a better visualization of the sizing-matrix data, including parameter variation,
changes in weight, and design constraints, the data was used to construct a carpet plot (Raymer,
2012, pp. 739–741). The carpet plot displays takeoff weight vs. wing loading for different T/W
values. In order to avoid clutter, the horizontal axis (i.e., W/S) was shifted to the left by three
units for each subsequently lower T/W. In other words, for a T/W of 0.26, the W/S retained its
original vertical position, but was shifted by three or six units to the left for T/W values of 0.22
and 0.14, respectively (figure 2).
Fig. 2. Carpet plot.

5. 5
In a carpet plot, the optimal aircraft is found visually as the plot’s lowest point that meets all
constraints (here, takeoff distance and rate of climb). Contrary to expectation, this did not occur
at the intersection of the two constraint curves. Rather, the rate of climb was dominant in
dictating the minimum takeoff weight. As the constraint curves do not intersect, the highest wing
loading investigated (W/S = 65 lb/ft2
) was selected as the parameter on whose curve to identify
the point of optimal (i.e., lowest) weight (visualized by an open red circle). This point
corresponds to a takeoff gross weight of 74,500 lb. However, as engine choice is not a
continuous variable, the minimum weight realizable with the engine providing sufficient thrust
(that is, the GE CF34-8C5, which corresponds to a T/W of 0.22) is 75,150 lb. Compared to our
previous refined weight estimate of W0 = 76,068 lb, the carpet-plot optimization analysis brought
the weight further down by 1568 or 918 lb, respectively. It would be necessary to perform
mathematically more sophisticated and complex optimization methods in order to hone in on a
further improved optimal weight estimate.
Acknowledgment
I would like to thank my team member Eric Nachtigall for his help with developing the
MATLAB code for conducting the iterative refined weight estimate.

6. 6
References
Anderson, John D. Jr. Aircraft Performance and Design. New York: McGraw Hill, 1999.
“CF34-3 Turbofan Engine.” Web. Accessed 9/1/2016.
http://www.geaviation.com/engines/docs/commercial/datasheet-CF34-3.pdf
“CF34-10A Turbofan.” Web. Accessed 9/1/2016.
http://www.geaviation.com/engines/docs/commercial/datasheet-CF34-10A.pdf
“Gulfstream G650ER.” Web. Accessed 9/1/2016.
http://www.gulfstream.com/images/uploads/brochures/aircraft/G650ERSpecSheet.pdf
Raymer, Daniel P. Aircraft Design: A Conceptual Approach. Reston: American Institute of
Aeronautics and Astronautics, Inc., 2012.

7. 7
Appendix
MATLAB code used for refined weight estimate. This code snippet was iterated until
convergence of the computed variables was attained. It is part of the complete MATLAB
program submitted with our team’s report.

8. 8
Calculate Wing Loading
WS_stall = (1/2*rho_SL*v_stall^2*CL_max)/.97;
WS_cruise = q_cruise*sqrt(C_D0/(3*K));
WS_loiter = q_loiter*sqrt(C_D0/K);
WS = min([WS_stall,WS_cruise, WS_loiter]);
% K = (1/2*rho_cruise*v_cruise^2/WS)^2*C_D0/3;
% AR = 1/(pi*e*K);
K = 1/(pi*e*AR);
CD = C_D0+K*CL_max^2;
CL_max_to_CD = CL_max/CD;
Calculate Thrust to Weight Ratio
v_climb = sqrt(2/rho_cruise*sqrt(K/C_D0)*WS);
TW_climb = (RC/v_climb + 2*sqrt(K*C_D0))*.97;
TW_takeoff = WS/(TOP*1*CL_max/1.21);
TW = max([TW_takeoff, TW_climb, TW_climb]);
Estimate weight
f_cruise = (1 - c_t*R/2*(rho_cruise*S/2/W_midcruise)^.5*4/3*(3*K*C_D0^3)^.25)^2;
f_loiter = exp(-c_t*E*2*sqrt(K*C_D0));
f_f = 1.06*(1 - .97*.985*f_loiter*f_cruise*.995);
%W_fuel = f_f*W_midcruise;
W_fuel = 40000;
S = W_midcruise/(WS);
b = sqrt(AR*S);
c_avg = S/b; %chord length
taper_tail = 0.5;
% Refined weight estimate - correlations from Raymer chapter 15
S_tailH = 0.7*c_avg*S/(L_plane - x_w); % 0.7 recommended from Raymer
S_tailV = 0.04*b*S/(L_plane - x_w); % 0.04 recommended from Raymer
W_wings = .036*S^.758*W_fuel^.0035*(AR/cosd(sweep)^2)^.6* ...
q_cruise^.006*taper^.04*(100*.13/cosd(sweep))^-.3*(1.5*N*W_midcruise)^0.49;
W_tailH = .016*(1.5*N*W_midcruise)^.414*q_cruise^.168*S_tailH^.896*(100*t_tail/...
cos(sweep))^-.12*(AR/cosd(sweep)^2)^.043*taper_tail^-.02;
W_tailV = .073*(1.5*N*W_midcruise)^.376*q_cruise^.122*S_tailV^.873*(100*t_tail/...
cos(sweep))^-.49*(AR/cosd(sweep)^2)^.357*taper_tail^.039;
W_fuse = .052*SA_fuse^1.086*(1.5*N*W_midcruise)^0.177*L_tail^-.051* ...
(1/sqrt(4*K*C_D0))^-0.072*q_cruise^.241;
W_gear = 0.057*W_midcruise;
W_engines = 2.575*2*W_engine^0.922;
W_etc = 0.1*W_midcruise;
W_empty = W_wings+W_tailH+W_tailV+W_fuse+W_gear+W_engines+W_etc;
W_0 = W_fuel + W_payload + W_empty;
plane_weights(i) = W_0;