Library / Flight dynamics and control / FLT-04
Flight dynamics and control
Specific total energy rate
Rate of change of specific total energy, from the climb rate and the along-track acceleration.
- Ref
- FLT-04
- Rev
- v1
- Field
- Flight dynamics and control
- Result
- ėT [W/kg]
Differentiating eT = gh + Va²/2 gives ėT = gḣ + VaV̇a. The first term is the rate at which height is gained, the second the rate at which speed is gained, each per unit mass. In total-energy control the throttle sets the sum, and the elevator distributes it between climb and acceleration. The result is in W/kg.
Rate of change of specific total energy, from the climb rate and the along-track acceleration. Thrust sets the total, and the elevator moves energy between height and speed.
Calculator
- Climb part, gḣ19.62 W/kg
- Speed part, VaV̇a6 W/kg
- Climb rate that the total could support at constant speed, ėT/g2.612 m/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Of 25.62 W/kg, 19.62 goes into height and 6 into speed. Held at constant speed, the same 2 m/s climb would need only 19.62 W/kg; the other 6 W/kg is the cost of accelerating at the same time.
- The energy rate equals the net power per unit mass, (T − D)Va/m for small angle of attack, where T is thrust and D drag. At 25.62 W/kg a 1000 kg aircraft needs about 25.62 kW of surplus power over drag.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W/kg) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ḣ Climb rate | −5 to 10 m/s | −43.05 | 104.1 | 147.2 | +0.3 m/s | +11.5 % |
| V̇a Airspeed rate of change | −1 to 1 m/s² | −40.38 | 79.62 | 120 | +0.04 m/s² | +9.37 % |
| Va Airspeed | 20 to 300 m/s | 21.62 | 49.62 | 28 | +10% | +2.34 % |
Sweep of Climb rate
| Climb rate (m/s) | Specific total energy rate (W/kg) |
|---|---|
| −5 | −43.05 |
| −3.125 | −24.66 |
| −1.25 | −6.263 |
| 0.625 | 12.13 |
| 2.5 | 30.53 |
| 4.375 | 48.92 |
| 6.25 | 67.31 |
| 8.125 | 85.71 |
| 10 | 104.1 |
Sweep one variable
Variables and how each one contributes
Specific total energy rate
What the equation gives you. Everything else feeds it.
Climb rate
The climb rate ḣ in m/s, positive up and negative in descent. Directly adds the climb term gḣ to the total: each extra metre per second of climb needs 9.81 W/kg. Raising it by 1 m/s (to 3 m/s) moves the result by +38.3 %, from 25.62 to 35.43 W/kg.
Airspeed
The airspeed in m/s. Multiplies the acceleration term: the same acceleration takes more energy at higher speed. Raising it by 10 % (to 66) moves the result by +2.34 %, from 25.62 to 26.22 W/kg.
Airspeed rate of change
The airspeed rate of change in m/s², positive when accelerating. Directly adds the speed term VaV̇a to the total. Raising it by 0.1 m/s² (to 0.2 m/s²) moves the result by +23.4 %, from 25.62 to 31.62 W/kg.
Worked example with the values above
- Climb part: gḣ = 9.81 × 2 = 19.62 W/kg.
- Speed part: VaV̇a = 60 × 0.1 = 6 W/kg.
- ėT = 19.62 + 6 = 25.62 W/kg.
Assumptions and limits
- Still air, constant gravity and a point-mass aircraft: the airspeed and the inertial speed are taken to be the same.
- Small flight-path angles where the speed along the path is close to the speed through the air.
Common mistakes
- Getting the sign of the climb rate wrong. Climb is positive here; descent is negative and removes potential energy.
- Expecting throttle to command altitude and elevator to command speed. In total-energy control, throttle sets the total energy rate and pitch distributes it between height and speed.
Statistics and uncertainty
Both terms are measured quantities with different noise: climb rate comes from differentiating barometric or GNSS altitude, and acceleration from differentiating airspeed. Differentiation amplifies noise, so these rates are usually filtered or built from inertial sensors. A 0.05 m/s² error in the acceleration gives an energy-rate error of 3 W/kg at 60 m/s.
Where it is used
- Total-energy control of throttle and pitch for fixed-wing aircraft.
- Checking whether a climb or acceleration is paid for by thrust or by giving up speed.
Maths you need
- Time derivative: the rate of change of a quantity
- Power: energy per unit time