Aircraft Takeoff Weight: Mission Fuel Fraction Sizing

Mission segments are multiplied as retained mass ratios $q_i=m_{i+1}/m_i$. For a jet, $q_R=\exp[-Rc_T/(V(L/D))]$ and $q_E=\exp[-Ec_{T,l}/(L/D)_l]$, with weight-flow TSFC in s⁻¹. For a propeller, replace the cruise coefficient by $g c_P/[\eta_p(L/D)]$ and the hold coefficient by $g c_{P,l}V_l/[\eta_{p,l}(L/D)_l]$, with mass-flow power SFC in kg/J. Constant-speed holds use the exponential; the constant-lift-coefficient propeller option uses $q_E=(1+k_E E/2)^{-2}$, obtained by integrating $V\propto\sqrt m$. Reserve cruise and hold form a separate segment. Loaded fuel is $(1+f_{extra})(1-\prod q_i)m_0$.

Empty fraction is $f_e=k_e A(m_0/m_{lb})^C$, where $m_{lb}=0.45359237$ kg. A fixed fraction is implemented with $C=0$. The numerator of the sizing equation is payload plus crew: $m_0=(m_{payload}+m_{crew})/(1-f_s-f_e)$. A bracketed Newton iteration solves this nonlinear balance and rejects a root outside the mass ceiling.

Battery flight holds mass constant. Cruise and hold use $gR/[\eta_d\eta_p(L/D)]$ and $gVE/[\eta_d\eta_p(L/D)]$ energy per gross mass. Climb adds potential energy and drag work along the climb path. Warm-up, descent and landing energy allowances are inputs. Battery fraction is $(1+f_{extra,E})e_{mission}/(u e_{pack})$. The constant-weight range equation follows Staack, Sobron and Krus (2021), Eq. 2; other segment energy equations are derived from work and power balance here.

For $D=1-f_s-(1+C)f_e$, payload growth is $1/D$ and range growth is $m_0(\partial f_s/\partial R)/D$. Range differentiation uses 0.01 km; parameter elasticities use a 0.01% perturbation. Each plotted design point is solved independently.

An independent adaptive ODE integration of mass depletion checks the closed-form hold equation, including the propeller constant-lift-coefficient case.

Payload and crew remain aboard. L/D, SFC, propeller efficiency and drive efficiency are constant within each segment. Jet TSFC is fuel weight flow divided by thrust; a mass-flow TSFC in kg/(N·s) must first be multiplied by standard gravity. Propeller SFC is fuel mass per shaft energy.

Range is total cruise distance, excluding the separately modeled climb, descent and reserve. A constant-lift-coefficient hold starts at the entered loiter speed each time. Electric climb is at constant speed; energy recovery in descent is zero. Pack specific energy includes battery enclosure and management hardware; usable fraction includes discharge and aging limits. Empty mass includes propulsion and aircraft systems but excludes crew, payload and the entire battery pack or loaded fuel.

In the segment table, storage draw fraction means fuel burned divided by loaded fuel mass, or battery energy used divided by nominal installed pack energy. Extra allowances remain outside the phase draw.

In the parameter sensitivity chart, the propulsion parameter is cruise/hold SFC for jet and propeller aircraft, or installed pack specific energy for battery electric aircraft.

First-estimate, fixed-wing conceptual sizing. The conventional aircraft coefficients are historical statistical fits, not component weight estimates; select or calibrate a custom fit for the aircraft class and weight scale. For battery aircraft, use a defensible custom correlation or fixed empty fraction excluding the pack. Technology multipliers do not prove structural feasibility.

As in the Raymer mission method, m₀ is the initial design gross mass before the combined warm-up / takeoff allowance. It is not the runway mass after taxi. The final mass shown is after the modeled reserve is flown; normal trip landing mass, with reserves retained, is available in the calculation steps. Reserve distance, hold and extra allowance are user design choices, not a regulatory fuel policy; avoid counting the same reserve twice.

Mass closure establishes only this model's accounting consistency. It does not establish wing area, installed thrust or power, climb performance, stability, CG, structural strength, stall speed, field length, pack power capability, cooling, battery fire protection, usable tank volume or certified limits. Do not use the tool as an operational loading or dispatch calculator. Check a developed design with component buildup and a detailed mission analysis.

The solve is bounded by the entered mass ceiling and relative closure tolerance. Gaps in trade curves are rejected designs; they are not interpolated feasible points. All curves have 81–101 independently solved samples.