Mission analysis

The previous eleven lessons each answered one question about the Tern: how much drag it makes, how fast it climbs, how far it glides, how much runway it needs. An airline does not buy answers to separate questions. It buys an aircraft that flies a specific mission, from gate to gate, with a given payload, on a given route, every day, within the rules. Mission analysis puts all the pieces together. This final lesson builds the Tern's mission from Wichita to Denver, accounts for fuel and battery energy segment by segment, adds the reserves the rules require, and uses the result for the trade studies that shape both the airline's operation and the aircraft's design.

The mission profile

A mission profile breaks a flight into segments, each flown in a known way. A typical commuter mission has taxi out, take-off and initial climb, climb to cruise altitude, cruise, descent, approach and landing, and taxi in. Each segment is analyzed with the tools of earlier lessons: take-off with the ground roll of Lesson 9, climb with the excess power of Lesson 5, cruise with the specific air range of Lesson 7, descent with the gliding performance of Lesson 6 and landing with Lesson 10. The mission analysis adds up the time, distance, fuel and energy for each segment.

Hybrid-electric aircraft add a second energy account. The Tern carries fuel for its turbine generator and a battery that supplies peak power. The analysis must track both, and it must respect the rules Arjun set in Lesson 5: never let usable battery energy fall below 40 kWh, so a full-power go-around and climb remain available at any point. The battery also enables two tricks that conventional turboprops cannot use: taxiing on battery power alone, which saves fuel and noise at the gate, and recovering a little energy in the descent by windmilling the propellers.

Reserves

The fuel a flight must carry has several parts. The trip fuel covers the planned flight from engine start to shutdown. Contingency fuel, often 5 percent of the trip fuel or a fixed minimum, covers forecast errors, routing changes and other variations. Alternate fuel covers a missed approach at the destination, a climb and a cruise to the alternate airport and a landing there. The final reserve is typically 30 or 45 minutes of holding, depending on the operation. Operators may carry extra fuel at the captain's discretion for weather or delays. The precise requirements are set out in FAA operating rules (14 CFR Parts 91 and 135); the analysis here uses representative values that the airline's planners will refine.

# Mission analysis: fuel and battery energy for the Tern on the launch route, with reserves and a speed trade study
import math
g, S, CD0, k = 9.81, 28.0, 0.028, 0.039
OEW, PAYLOAD, ROUTE = 4200.0, 1100.0, 680      # kg, kg, km
SFC = 0.32                                      # kg of fuel per kWh of generator shaft output
USABLE, BAT_RES = 120.0, 40.0                   # usable battery energy and minimum battery reserve, kWh
def rho_at(h):
    T = 288.15 - 0.0065 * h
    return 101325 * (T / 288.15) ** 5.2559 / (287.05 * T)
def eta(V): return 0.84 * (1 - math.exp(-V / 30))
def shaft_power(m, V, rho):                     # shaft power for level flight, kW
    q = 0.5 * rho * V**2; CL = m * g / (q * S)
    return q * S * (CD0 + k * CL**2) * V / eta(V) / 1000
def mission(cruise_kmh, fuel_load=600.0, verbose=False):
    fuel, bat, dist, t, log = 0.0, USABLE, 0.0, 0.0, []
    def seg(name, minutes, gen_kw, bat_kw, km):
        nonlocal fuel, bat, dist, t
        f = SFC * gen_kw * minutes / 60; e = bat_kw * minutes / 60 / 0.95 if bat_kw > 0 else bat_kw * minutes / 60 * 0.9
        fuel += f; bat = min(USABLE, bat - e); dist += km; t += minutes
        log.append(f"  {name:22s} {minutes:5.1f} min {f:6.1f} kg  battery {-e:+6.1f} kWh  {km:5.0f} km")
    m0 = OEW + PAYLOAD + fuel_load              # take-off mass with the planned fuel load
    seg("Taxi out (battery)", 8, 0, 30, 0)
    seg("Take-off, initial climb", 2, 760, 140, 6)
    seg("Climb to 3000 m", 8.2, 710, 190, 34)
    rho = rho_at(3000); V = cruise_kmh / 3.6
    descent_km, approach_km = 65, 20
    cruise_km = ROUTE - dist - descent_km - approach_km
    m, x, cf, ct, recharge = m0 - fuel, 0.0, 0.0, 0.0, 0.0
    while x < cruise_km:                         # 5 km steps through the cruise
        P = shaft_power(m, V, rho)
        spare = max(0.0, 665 - P)                # spare generator power charges the battery
        charge = min(spare, 100.0) if bat + recharge < USABLE else 0.0
        dt = 5 / cruise_kmh
        f = SFC * (P + charge) * dt
        cf += f; m -= f; ct += dt; recharge += charge * dt * 0.9; x += 5
    fuel += cf; bat = min(USABLE, bat + recharge); dist += cruise_km; t += ct * 60
    log.append(f"  {'Cruise at ' + str(cruise_kmh) + ' km/h':22s} {ct*60:5.1f} min {cf:6.1f} kg  battery {recharge:+6.1f} kWh  {cruise_km:5.0f} km")
    seg("Descent (regeneration)", 12, 150, -20, descent_km)
    seg("Approach and landing", 6, 280, 20, approach_km)
    seg("Taxi in (battery)", 5, 0, 30, 0)
    trip = fuel
    contingency = 0.05 * trip
    alternate = 150 / 1.84 + 20                  # 150 km at the cruise specific air range, plus climb allowance
    final_reserve = 82.0                         # 45 minutes holding at 1500 m (Lesson 8)
    total = trip + contingency + alternate + final_reserve
    if verbose:
        print(f"Mission: {ROUTE} km, payload {PAYLOAD:.0f} kg")
        print("\n".join(log))
        print(f"Trip fuel {trip:.0f} kg; contingency {contingency:.0f} kg; alternate {alternate:.0f} kg; final reserve {final_reserve:.0f} kg")
        print(f"Fuel required {total:.0f} kg; take-off mass {OEW + PAYLOAD + total:.0f} kg (limit 6500)")
        print(f"Battery on arrival {bat:.0f} kWh (minimum {BAT_RES:.0f}); block time {t:.0f} min")
    return t, trip, bat, total
fuel = 600.0
for _ in range(3):                               # the fuel load affects the mass, so repeat until it settles
    fuel = mission(380, fuel)[3]
mission(380, fuel, verbose=True)
print("Cruise speed trade study:")
for v in (300, 330, 360, 380):
    t, trip, bat, total = mission(v, fuel)
    print(f"  {v} km/h: block {t:4.0f} min, trip fuel {trip:4.0f} kg, battery on arrival {bat:4.0f} kWh")

Mission: 680 km, payload 1100 kg
  Taxi out (battery)       8.0 min    0.0 kg  battery   -4.2 kWh      0 km
  Take-off, initial climb   2.0 min    8.1 kg  battery   -4.9 kWh      6 km
  Climb to 3000 m          8.2 min   31.1 kg  battery  -27.3 kWh     34 km
  Cruise at 380 km/h      87.6 min  305.0 kg  battery  +36.8 kWh    555 km
  Descent (regeneration)  12.0 min    9.6 kg  battery   +3.6 kWh     65 km
  Approach and landing     6.0 min    9.0 kg  battery   -2.1 kWh     20 km
  Taxi in (battery)        5.0 min    0.0 kg  battery   -2.6 kWh      0 km
Trip fuel 363 kg; contingency 18 kg; alternate 102 kg; final reserve 82 kg
Fuel required 564 kg; take-off mass 5864 kg (limit 6500)
Battery on arrival 115 kWh (minimum 40); block time 129 min
Cruise speed trade study:
  300 km/h: block  152 min, trip fuel  312 kg, battery on arrival  115 kWh
  330 km/h: block  142 min, trip fuel  325 kg, battery on arrival  115 kWh
  360 km/h: block  134 min, trip fuel  346 kg, battery on arrival  115 kWh
  380 km/h: block  129 min, trip fuel  363 kg, battery on arrival  115 kWh

The launch route

The program flies the 680 km route to Denver with 1100 kg of payload, nine passengers and their bags plus some freight. Taxi out on battery uses about 4 kWh. The take-off and climb to 3000 m use 39 kg of fuel and about 32 kWh of battery energy. The cruise at 380 km/h covers 555 km in 88 minutes on 305 kg of fuel, and the small surplus of generator power at the lighter cruise weights recharges the battery. The descent and approach use little fuel. The trip takes 129 minutes from gate to gate and burns 363 kg of fuel.

Adding 18 kg of contingency, 102 kg for an alternate and 82 kg of final reserve gives a fuel requirement of 564 kg and a take-off mass of 5864 kg, well inside the 6500 kg maximum. The battery arrives with about 115 kWh, far above its 40 kWh floor. The mission fits easily, with over 600 kg of take-off weight to spare. That spare capacity matters at Denver in particular: the airport sits about 1650 m(5430 ft) above sea level, and on a hot summer afternoon the Tern's take-off from there for the return flight, as Lesson 9 showed for a 1600 m airport, needs about a quarter more runway than at sea level.

Trade studies

Once the mission model exists, it can answer questions quickly. The most common is cruise speed. Flying faster shortens the block time, which lets the airline fly more trips per day with the same aircraft and crew and pleases passengers; flying slower saves fuel. The trade study in the program shows the balance.

Cruising at 380 km/h rather than 300 km/h saves 23 minutes per trip for 51 kg of extra fuel. Whether that is worth it depends on the airline's costs. If the aircraft's hourly ownership and crew costs are high compared with fuel, as they usually are for small aircraft, the faster speed wins. If fuel prices spike, or a carbon charge raises the cost of each kilogram, the balance shifts toward the slower speed. Airlines often use a cost index, the ratio of time-related to fuel-related costs, to let the flight management system choose the speed automatically.

Trades that reach back into the design

Mission analysis is not only for operations. Early in a design, the same model is run thousands of times to size the aircraft. Each design choice changes the mission: a larger wing lowers the stall speed and shortens the field lengths but adds weight and drag in cruise; a bigger battery adds climb power and reserve energy but costs payload; a larger generator raises the cruise speed but burns more fuel at part power. Arjun's team at Prairiewind Aerospace uses the mission model to test each change against the airline's requirements, keeping a matrix of the results.

These trades show why drag reduction is so valuable: it improves every cruise-related measure at once, with no penalty elsewhere. They also show why the field-length requirement at the shortest strips and the cruise-speed requirement pull the wing size in opposite directions. The final design is a balance that meets every requirement with a margin, rather than an optimum for any one of them.

The Skiff's mission

The same method applies to the Skiff cargo drone. Its typical mission is a medical delivery: take off vertically from a hospital rooftop, transition to wing-borne flight, cruise to a clinic, descend, land vertically, and return. The hover segments dominate its energy budget, because hover power is several times cruise power for a small vehicle, so the mission model for the Skiff-W counts seconds of hover carefully. Its reserve rules are set by the operator's approved procedures rather than airline fuel rules, but they serve the same purpose: always enough energy to reach a safe landing site, whatever happens. With the endurance from Lesson 8, the Skiff-W comfortably covers a round trip of more than 100 km with a reserve.

Bringing it together

Aircraft performance is the engineering of what an aircraft can do with the forces acting on it. Lift, drag, thrust and weight, combined through Newton's laws and energy, give the speeds, distances, rates and endurance that define an aircraft's capability. Certification and operating rules then turn capability into the margins that make flying safe. Mission analysis brings it all together into the answer the customer cares about: can this aircraft carry this load on this route, reliably, at an acceptable cost? For the Tern on its launch route, the answer is yes, with room to grow.