System-level Model

The objective of the optimization problem presented in this work is to minimize fuel consumption, or equivalently fuel weight, \(W_{fuel}\), using an adaptation of the Breguet range formulation introduced in [Hoburg, 2013]. The purpose of the system-level model is threefold: it enforces system-level performance constraints such as required range and minimum cruise speed, it encodes weight and drag buildups, and it constrains system-level properties such as the aircraft’s and moment of inertia. In doing these things, it also couples the subsystem models.

Model Assumptions

The model presented in this work is a set of constraints that describe the performance and design of a conventional-configuration narrowbody aircraft, with a simple cruise-only mission profile. A more sophisticated mission profile is left for future work.

Model Description

Variable tables are available for download below:

• Free variables
• Fixed variables

Flight Performance

The Breguet range formulation is discretized over multiple cruise segments to improve accuracy, meaning the constraints from [Hoburg, 2013] apply during each of the \(N\) flight segments. The \(n\) subscript is used to represent the \(n^{th}\) flight segment where \(n=1...N\). For readability, these subscripts are not used in the remainder of the manuscript, but still apply.

\[\begin{split}\begin{aligned} \sum_{n=1}^{N} R_{n} &\geq R_{req} \\ R_{n+1} &= R_{n} \\ R_{n} &\leq \frac{V_{\infty_{n}}}{n_{eng}c_{T_{n}} g} \frac{W_{{avg}_{n}}}{D_{n}} z_{bre_{n}}\\ W_{fuel_{n}} &\geq \left(z_{bre_{n}} + \frac{z_{bre_{n}}^2}{2} + \frac{z_{bre_{n}}^{3}}{6} \right) W_{end_{n}} \\ W_{fuel_{n}} &\geq n_{eng} {c_{T_{n}}} D_{n} t_{n} \\ \sum_{n=1}^{N}W_{fuel_{n}} &\leq W_{f_{primary}} \\ V_{\infty_{n}} t_{n} &= R_{n} \\ W_{start_{n}} &\geq W_{end_{n}} + W_{fuel_{n}} \\ W_{start_{n+1}} &= W_{end_{n}} \\ W &\geq W_{dry} + W_{payload} + f_{fuel_{res}} W_{f_{primary}} \\ W_{start_{0}} &= W \\ W_{end_{N}} &\geq W_{dry} + W_{payload} + f_{fuel_{res}} W_{f_{primary}}\\ W_{avg_{n}} &\geq \sqrt{W_{start_{n}} W_{end_{N}}} + W_{buoy_{n}} \\ \left(\frac{L}{D}\right)_{n} &= \frac{W_{avg_{n}}}{D_{n}} \end{aligned}\end{split}\]

In the remainder of this manuscript, \(W\) refers to the corresponding flight segment’s \(W_{avg}\).

The dry weight and drag of the aircraft are constrained using simple buildups of each component’s weight and drag.

\[\begin{split}\begin{aligned} W_{dry} &\geq W_{wing} + W_{fuse} + W_{vt} + W_{ht} + W_{lg} + W_{eng} + W_{misc} \\ D_n &\geq D_{wing_n} + D_{fuse_n} + D_{vt_n} + D_{ht_n}\end{aligned}\end{split}\]

Mach number is constrained to be greater than a user-specified minimum value.

\[\begin{split}\begin{aligned} M &= \frac{V_{\infty}}{a} \\ M &\geq M_{min}\end{aligned}\end{split}\]

The takeoff model is taken directly from [Hoburg, 2013]. An additional constraint on takeoff velocity is added to ensure adequate margin above stall speed [Anderson, 2001].

\[\begin{split}\begin{aligned} {x_{TO}} &\leq {l_r} \\ 1 + {y} &\leq 2\frac{ {g} {x_{TO}}{T_e}}{{V_{TO}}^{2} {W}} \\ 1 &\geq 0.0464\frac{{\xi}^{2.7}}{{y}^{2.9}} + \frac{{\xi}^{0.3}}{{y}^{0.049}}\\ {\xi} &\geq \frac12 \frac{{\rho_{TO}}{V_{TO}}^{2} {S_w}{C_D}}{{T_e}} \\ {V_{TO}} &= 1.2\sqrt{\frac{2{W}}{C_{L_{w,max}}} {S_w} {\rho_{TO}}} \end{aligned}\end{split}\]

Atmospheric pressure, density, temperature, and speed of sound are constrained using the atmosphere model described in [York, 2017]. Dynamic viscosity is constrained using the viscosity model developed in [Kirschen, 2016].

System-level Properties

The constraint for the aircraft is -compatible, and is satisfied during each flight segment. The fuselage and payload weights are assumed to be evenly distributed through the length of the fuselage, and the wing weight acts directly at its area centroid, \(x_{wing} + \Delta x_{ac_w}\). It is assumed that the fuel weight shifts in proportion to the remaining fuel fraction, \(f_{fuel}\), and that a reserve fuel fraction, \(f_{fuel_{res}}\), remains in the wing. The wingbox forward bulkhead location, \(x_b\), is used as a surrogate variable for engine .

\[\begin{split}\begin{aligned} W x_{CG_{n}} &\geq W_{wing} \left(x_{wing} + \Delta x_{ac_w}\right) + W_{f_{primary}} \left(f_{fuel_{n}} + f_{fuel_{res}}\right) \left(x_{wing} + \Delta x_{ac_w} f_{fuel_{n}}\right) \\ & +\frac{1}{2} \left(W_{fuse} + W_{payload}\right) l_{fuse} + W_{ht} x_{CG_{ht}} + \left(W_{vt} + W_{cone} \right) x_{CG_{vt}} \nonumber \\ & + n_{eng} W_{eng} x_b + W_{lg} x_{lg} + W_{misc} x_{misc} \nonumber\end{aligned}\end{split}\]

In the prior constraint, \(f_{fuel}\) is the percent of primary fuel remaining. \(f_{fuel}\) is represented adequately by a posynomial inequality since it has downward pressure.

\[f_{fuel_{n}} \geq \frac{\sum_{n=1}^{n}W_{fuel_{n}}}{W_{f_{primary}}}\]

The landing gear is constrained by the moment of each set of landing gear about the nose of the aircraft.

\[W_{lg} x_{lg} \geq W_{mg} x_m + W_{ng} x_n\]

The miscellaneous equipment includes only power systems in the current model, but is defined to allow for refinements in CG modeling in future work.

\[\begin{aligned} W_{misc} x_{misc} &\geq W_{hpesys} x_{hpesys}\end{aligned}\]

The aircraft’s moment of inertia is the sum of the inertias of its components.

\[\label{e:Iz_sum} I_z \geq I_{z_{wing}} + I_{z_{fuse}} + I_{z_{tail}}\]

The wing moment of inertia model includes the moment of inertia of the fuel systems and engines. It assumes that the wing and fuel weight are evenly distributed on the planform of the wing. This is an overestimate of the wing moment of inertia with full fuel tanks.

\[\label{e:Iz_wing} I_{z_{wing}} \geq \frac{n_{eng} W_{engine} y_{eng}^2}{g} + \left(\frac{W_{fuel_{wing}} + W_{wing}}{g}\right) \frac{{b_{w}}^3 c_{root_{w}}}{16 S_{w}} \left(\lambda_w + \frac{1}{3}\right)\]

The fuselage moment of inertia includes the payload moment of inertia. It is assumed that payload and fuselage weight are evenly distributed along the length of the fuselage. The wing root quarter-chord location acts as a surrogate for the of the aircraft.

\[I_{z_{fuse}} \geq \left(\frac{W_{fuse} + W_{pay}}{g}\right) \left(\frac{x_{wing}^3 + l_{vt}^3}{3l_{fuse}}\right)\]

The moment of inertia of the tail is constrained by treating the tail as a point mass.

\[\label{e:Iz_tail} I_{z_{tail}} \geq \left(\frac{W_{apu} + W_{tail}}{g}\right) l_{vt}^2\]