Range Equation

Info

The relation describing the specific air range of an aircraft in cruise can be adapted to calculate the fuel required for a given range \(R\). The governing equation for the range is:

\[ R = \frac{1}{g} \int_{m_2}^{m_1} \frac{V\cdot L/D}{TSFC \cdot m} \text{d}m \]

Depending on which variables are assumed constant during the flight, different analytical solutions to the integral can be derived:

Variables assumed Constant	Variables allowed to Vary	Additional Assumptions	Solution	Function
\(TSFC\) \(L/D\) \(V\)	\(m\)	none	"Breguet range equation (climb-cruise)"	`calculate_fuel_consumption_breguet` `calculate_fuel_consumption_breguet_improved`
\(TSFC\) \(h\) \(V\)	\(m\) \(L/D\)	low-speed drag polar applies \(C_D = C_{D_0} + KC_L^2\)	"arctan solution (step-climb)"	`calculate_fuel_consumption_stepclimb_arctan`
\(TSFC\) \(h\) \(V\)	\(m\) \(L/D\)	none	"numerical solution (step-climb)"	`calculate_fuel_consumption_stepclimb_integration`

References

Section 13.2 in Young, T. M. (2018). Performance of the Jet Transport Airplane (Section 13.7.3 "Fuel Required for Specified Range"). John Wiley & Sons. doi:10.1002/9781118534786

Note also that some authors spell "Breguet" as "Bréguet" (with accent on the "e"). This is not limited to non-French speakers. Even in his original 1923 publication "Calcul du Poids de Combustible Consummé par un Avion en vol Ascendant" , Breguet's name is spelled without an accent:

But in the same publication, an aircraft manufactured by his company is spelled with accent:

jetfuelburn.rangeequation ¶

calculate_fuel_consumption_breguet ¶

calculate_fuel_consumption_breguet(
    R, LD, m_after_cruise, V, TSFC
)

Given a flight distance (=range) \(R\) and basic aircraft performance parameters (see table), returns the fuel mass burned during the flight \(m_f\) [kg] based on a flight schedule where the lift-coefficient and velocity are constant during cruise ("cruise-climb"). This solution is also known as the Breguet range equation.

Fuel mass is calculated as:

\[ m_f = (e^{\frac{R \cdot TSFC \cdot g}{L/D \cdot v}} - 1 ) m_2 \]

where:

Symbol	Dimension	Description
\(m_{fuel}\)	[mass]	Fuel required for the mission of the aircraft
\(R\)	[distance]	Range of the aircraft (=mission distance)
\(TSFC\)	[time/distance]	Thrust Specific Fuel Consumption of the aircraft during cruise
\(g\)	[acceleration]	Acceleration due to gravity
\(L/D\)	[dimensionless]	Lift-to-Drag ratio of the aircraft
\(v\)	[speed]	Average cruise speed of the aircraft (TAS)
\(m_2\)	[mass]	Mass of the aircraft after cruise segment

Note that thrust-specific fuel consumption (TSFC) is of dimension [time/distance], as the commonly used unit [g/kNs] simplifies accordingly.

Notes

Key assumptions of this fuel calculation function:

Parameter	Assumption
data availability	adequate for both current aircraft (reports) and future aircraft (studies)
aircraft payload	variable
climb/descent	not considered, only cruise-phase
fuel reserves	can be considered through \(m_2\)
alternate airport	can be considered through \(m_2\)

Warnings

This analytical fuel calculation method represents a simplification of actual flight dynamics. Specifically, this function assumes a flight schedule with constant airspeed and constant lift coefficient, resulting in a cruise-climb. Young (2017) shows this in Section 13.3.3 "Second Flight Schedule":

(...) the airplane must be flown in a way that will ensure that the ratio (...) [of weight to air density] remains constant. This is possible if the airplane is allowed to climb very slowly so that the relative density (...) decreases in direct proportion to the decrease in airplane weight (...).

References

Young, T. M. (2018). Performance of the Jet Transport Airplane (Section 13.7.3 "Fuel Required for Specified Range"). John Wiley & Sons. doi:10.1002/9781118534786
Cavcar, M. (2006). Bréguet range equation?. Journal of Aircraft. doi:10.2514/1.17696
Range (Aeronautics) entry on Wikipedia

calculate_fuel_consumption_breguet_improved ¶

calculate_fuel_consumption_breguet_improved(
    R,
    LD,
    m_after_cruise,
    V,
    V_headwind,
    TSFC,
    lost_fuel_fraction=0.0152,
    recovered_fuel_fraction=0.001,
)

Given a flight distance (=range) \(R\) and basic aircraft performance parameters (see table), returns the fuel mass burned during the flight \(m_f\) [kg] based on a flight schedule where the lift-coefficient and velocity are constant during cruise ("cruise-climb"), while taking into account headwind effects and fuel losses during takeoff and climb, as well as fuel recovery during descent and landing.

Fuel mass is calculated as:

\[ m_{\text{fuel}} = m_{\text{LDG}} \left( \frac{1}{\exp\left\{ \frac{-R}{H \left(1 - \frac{V_{\text{HW}}}{V}\right)} \right\} - \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} + \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}}} - 1 \right) \]

with

\[ H = \frac{L/D \cdot V}{TSFC \cdot g} \]

where:

Symbol	Dimension	Description
\(m_{f}\)	[mass]	Fuel required for the mission of the aircraft
\(m_{LDG}\)	[mass]	Mass of the aircraft after cruise segment
\(R\)	[distance]	Range of the aircraft (=mission distance)
\(TSFC\)	[time/distance]	Thrust Specific Fuel Consumption of the aircraft during cruise
\(g\)	[acceleration]	Acceleration due to gravity
\(L/D\)	[dimensionless]	Lift-to-Drag ratio of the aircraft
\(V\)	[speed]	Average speed of the aircraft (TAS)
\(V_{HW}\)	[speed]	Average speed of headwind component
\(\frac{\Delta m_L}{m_{TO}}\)	[dimensionless]	Lost fuel fraction
\(\frac{\Delta m_R}{m_{TO}}\)	[dimensionless]	Recovered fuel fraction

Derivation

Equation 19 in Randle et al. (2011) expresses the fuel mass burned \(m_f\) during a flight of distance \(s\) (=range \(R\)) as a function of the takeoff mass \(m_{TO}\). However, in practical applications, the more relevant known parameter is the landing mass after cruise

\(m_{LDG}\) (= \(m_{after\_cruise}\))

Therefore, the equation is rearranged here to solve for \(m_f\) as a function of \(m_{LDG}\):

\[\begin{align} m_f &= m_{\text{TO}} \left( 1 - e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}} \right) \\ m_f &= (m_f + m_{\text{LDG}}) \left( 1 - e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}} \right) \\ m_f &= (m_f + m_{\text{LDG}}) \cdot 1 + (m_f + m_{\text{LDG}}) \left( -e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}} \right) \\ m_f &= m_f + m_{\text{LDG}} + (m_f + m_{\text{LDG}}) \left( -e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}} \right) \\ 0 &= m_{\text{LDG}} + (m_f + m_{\text{LDG}}) \left( -e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}} \right) \\ -m_{\text{LDG}} &= (m_f + m_{\text{LDG}}) \left( -e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}} \right) \\ \frac{-m_{\text{LDG}}}{-e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}}} &= m_f + m_{\text{LDG}} \\ m_f &= \frac{-m_{\text{LDG}}}{-e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}}} - m_{\text{LDG}} \\ m_f &= m_{\text{LDG}} \left( \frac{-1}{-e^{-\frac{s}{H}} + \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} - \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}}} - 1 \right) \\ m_f &= m_{\text{LDG}} \left( \frac{1}{e^{-\frac{s}{H}} - \frac{\Delta m_{\text{lost}}}{m_{\text{TO}}} + \frac{\Delta m_{\text{rec}}}{m_{\text{TO}}}} - 1 \right) \end{align}\]

Warnings

Unlike the equation in Randle et al. (2011), this function uses mass [kg] instead of weight [N] for the fuel.

Notes

Assumes a lost fuel fraction \(\Delta m_{lost}=0.0152\) and a recovered fuel fraction \(\Delta m_{rec}=0.001\) as per section D of Randle et al. (2011). Values can be adjusted through function parameters.

References

Eqn. 19 in Randle, W. E., Hall, C. A., & Vera-Morales, M. (2011). Improved range equation based on aircraft flight data. Journal of Aircraft. doi:10.2514/1.C031262

calculate_fuel_consumption_stepclimb_arctan ¶

calculate_fuel_consumption_stepclimb_arctan(
    R, h, K, C_D0, m_after_cruise, S, V, TSFC
)

Given a flight distance (=range) \(R\) and basic aircraft performance parameters (see table), returns the fuel mass burned during the flight \(m_f\) [kg] based on a flight schedule where aircraft velocity and altitude are constant during cruise.

Fuel mass is calculated as:

\[ m_f = \frac{(B + m_{LDG}^2) \tan(\theta)}{\sqrt{B} - m_{LDG} \tan(\theta)} \]

with

\[\begin{align*} \theta &= \frac{R \cdot g \cdot TSFC}{2 E_{max} V} \\ B &= \left( \frac{C_{D_0}}{K} \right) \left( \frac{\rho V^2 S}{2g} \right)^2 \\ E_{max} &= \frac{1}{2 \sqrt{C_{D_0} K}} \end{align*}\]

where:

Symbol	Dimension	Description
\(m_{fuel}\)	[mass]	Fuel required for the mission of the aircraft
\(R\)	[distance]	Range of the aircraft (=mission distance)
\(TSFC\)	[time/distance]	Thrust Specific Fuel Consumption of the aircraft during cruise
\(g\)	[acceleration]	Acceleration due to gravity
\(V\)	[speed]	Average cruise speed of the aircraft (TAS)
\(E_{max}\)	[energy/mass]	Maximum lift-to-drag ratio according to drag parabola
\(C_{D_0}\)	[dimensionless]	Zero-lift drag coefficient of the aircraft
\(K\)	[dimensionless]	Lift-dependent drag factor of the aircraft
\(\rho\)	[mass/volume]	Air density at cruise altitude
\(S\)	[area]	Wing reference area of the aircraft

Derivation

Eqn. 7.43 in Young (2018) expresses the range \(R\) of an aircraft as a function of the takeoff mass \(m_1\) and landing mass \(m_2\) after cruise, as well as the aircraft performance parameters. However, in practical applications, the fuel mass \(m_f\) should be calculated based on the landing mass after cruise \(m_2\).

Starting from the definition of aircraft masses:

\[\begin{align*} m_f &= m_1 - m_2 \\ m_1 &= m_f + m_2 \\ \end{align*}\]

The expression for range can be rearranged:

\[\begin{align*} R &= \frac{2E_{max}V}{g\bar{c}} \arctan\left\{ \frac{\sqrt{B_3}(m_1 - m_2)}{B_3 + m_1 m_2} \right\} \\ \Theta &:= \frac{R g \bar{c}}{2 E_{max} V} \\ \tan(\Theta) &= \frac{\sqrt{B_3}(m_1 - m_2)}{B_3 + m_1 m_2} \\ \tan(\Theta) &= \frac{\sqrt{B_3} m_f}{B_3 + (m_f + m_2)m_2} \\ \tan(\Theta)(B_3 + m_f m_2 + m_2^2) &= \sqrt{B_3} m_f \\ B_3 \tan(\Theta) + m_f m_2 \tan(\Theta) + m_2^2 \tan(\Theta) &= \sqrt{B_3} m_f \\ \tan(\Theta)(B_3 + m_2^2) &= m_f (\sqrt{B_3} - m_2 \tan(\Theta)) \\ m_f &= \frac{(B_3 + m_2^2) \tan(\Theta)}{\sqrt{B_3} - m_2 \tan(\Theta)} \\ \end{align*}\]

where:

\[\begin{equation*} B_3 = \left(\frac{C_{D_0}}{K}\right) \left(\frac{\rho V^2 S}{2g}\right)^2 \end{equation*}\]

References

Eqn. 7.43 and Eqn. 13.25a ff. in Young, T. M. (2018). Performance of the Jet Transport Airplane. John Wiley & Sons. doi:10.1002/9781118534786

Example

Editor (session: default) Run

import jetfuelburn
from jetfuelburn import ureg
from jetfuelburn.rangeequation import calculate_fuel_consumption_arctan
calculate_fuel_consumption_arctan(
    R=2000*ureg.nmi,
    h=35000*ureg.feet,
    K=0.045,
    C_D0=0.02,
    m_after_cruise=100*ureg.metric_ton,
    S=122.6*ureg.meter**2,
    V=800*ureg.kph,
    TSFC=17*(ureg.mg/ureg.N/ureg.s),
)

Output Clear

Source code in jetfuelburn/rangeequation.py

@ureg.check(
    "[length]",
    "[length]",
    "[]",
    "[]",
    "[mass]",
    "[area]",
    "[speed]",
    "[time]/[length]",  # [mg/Ns] = s/m
)
def calculate_fuel_consumption_stepclimb_arctan(
    R: pint.Quantity[float | int],
    h: pint.Quantity[float | int],
    K: float | int | pint.Quantity[float | int],
    C_D0: float | int | pint.Quantity[float | int],
    m_after_cruise: pint.Quantity[float | int],
    S: pint.Quantity[float | int],
    V: pint.Quantity[float | int],
    TSFC: pint.Quantity[float | int],
) -> float:
    r"""
    Given a flight distance (=range) $R$ and basic aircraft performance parameters (see table),
    returns the fuel mass burned during the flight $m_f$ [kg] based on a flight schedule where 
    aircraft velocity and altitude are constant during cruise.

    Fuel mass is calculated as:

    $$
        m_f = \frac{(B + m_{LDG}^2) \tan(\theta)}{\sqrt{B} - m_{LDG} \tan(\theta)}
    $$

    with

    \begin{align*}
        \theta &= \frac{R \cdot g \cdot TSFC}{2 E_{max} V} \\
        B &= \left( \frac{C_{D_0}}{K} \right) \left( \frac{\rho V^2 S}{2g} \right)^2 \\
        E_{max} &= \frac{1}{2 \sqrt{C_{D_0} K}}
    \end{align*}

    where:

    | Symbol     | Dimension         | Description                                                            |
    |------------|-------------------|------------------------------------------------------------------------|
    | $m_{fuel}$ | [mass]            | Fuel required for the mission of the aircraft                          |
    | $R$        | [distance]        | Range of the aircraft (=mission distance)                              |
    | $TSFC$     | [time/distance]   | Thrust Specific Fuel Consumption of the aircraft during cruise         |
    | $g$        | [acceleration]    | Acceleration due to gravity                                            |
    | $V$        | [speed]           | Average cruise speed of the aircraft (TAS)                             |
    | $E_{max}$  | [energy/mass]     | Maximum lift-to-drag ratio according to drag parabola                  |
    | $C_{D_0}$  | [dimensionless]   | Zero-lift drag coefficient of the aircraft                             |
    | $K$        | [dimensionless]   | Lift-dependent drag factor of the aircraft                             |
    | $\rho$     | [mass/volume]     | Air density at cruise altitude                                         |
    | $S$        | [area]            | Wing reference area of the aircraft                                    |


    ??? info "Derivation"

        Eqn. 7.43 in Young (2018) expresses the range $R$ of an aircraft as a function of the takeoff mass $m_1$ 
        and landing mass $m_2$ after cruise, as well as the aircraft performance parameters. 
        However, in practical applications, the fuel mass $m_f$ should be calculated based on the landing mass after cruise $m_2$.

        Starting from the definition of aircraft masses:

        \begin{align*}
        m_f &= m_1 - m_2 \\
        m_1 &= m_f + m_2 \\
        \end{align*}

        The expression for range can be rearranged:

        \begin{align*}
        R &= \frac{2E_{max}V}{g\bar{c}} \arctan\left\{ \frac{\sqrt{B_3}(m_1 - m_2)}{B_3 + m_1 m_2} \right\} \\
        \Theta &:= \frac{R g \bar{c}}{2 E_{max} V} \\
        \tan(\Theta) &= \frac{\sqrt{B_3}(m_1 - m_2)}{B_3 + m_1 m_2} \\
        \tan(\Theta) &= \frac{\sqrt{B_3} m_f}{B_3 + (m_f + m_2)m_2} \\
        \tan(\Theta)(B_3 + m_f m_2 + m_2^2) &= \sqrt{B_3} m_f \\
        B_3 \tan(\Theta) + m_f m_2 \tan(\Theta) + m_2^2 \tan(\Theta) &= \sqrt{B_3} m_f \\
        \tan(\Theta)(B_3 + m_2^2) &= m_f (\sqrt{B_3} - m_2 \tan(\Theta)) \\
        m_f &= \frac{(B_3 + m_2^2) \tan(\Theta)}{\sqrt{B_3} - m_2 \tan(\Theta)} \\
        \end{align*}

        where:

        \begin{equation*}
        B_3 = \left(\frac{C_{D_0}}{K}\right) \left(\frac{\rho V^2 S}{2g}\right)^2
        \end{equation*}


    References
    ----------
    Eqn. 7.43 and Eqn. 13.25a ff. in 
    Young, T. M. (2018). 
    Performance of the Jet Transport Airplane. 
    _John Wiley & Sons_. 
    doi:[10.1002/9781118534786](https://doi.org/10.1002/9781118534786)

    Example
    -------
    ```pyodide install='jetfuelburn'
    import jetfuelburn
    from jetfuelburn import ureg
    from jetfuelburn.rangeequation import calculate_fuel_consumption_arctan
    calculate_fuel_consumption_arctan(
        R=2000*ureg.nmi,
        h=35000*ureg.feet,
        K=0.045,
        C_D0=0.02,
        m_after_cruise=100*ureg.metric_ton,
        S=122.6*ureg.meter**2,
        V=800*ureg.kph,
        TSFC=17*(ureg.mg/ureg.N/ureg.s),
    )
    ```

    """
    if R == 0 * ureg.meter:
        return 0 * ureg.kg
    if R < 0 * ureg.meter:
        raise ValueError("Range must be greater than zero.")
    if h < 0 * ureg.meter:
        raise ValueError("Altitude must be greater than zero.")
    if m_after_cruise < 0 * ureg.kg:
        raise ValueError("Mass after cruise must be greater than zero.")
    if S <= 1 * ureg.meter**2:
        raise ValueError("Lift-to-Drag ratio must be greater than 1.")
    if V <= 0 * ureg.kph:
        raise ValueError("Cruise speed must be greater than zero.")
    if TSFC.magnitude <= 0:
        raise ValueError("Thrust Specific Fuel Consumption must be greater than zero.")

    E_max = 0.5 * (1 / math.sqrt(C_D0 * K))
    rho = _calculate_atmospheric_density(altitude=h)
    theta = (R * ureg.gravity * TSFC) / (2 * E_max * V)
    B = (C_D0 / K) * ((rho * V**2 * S) / (2 * ureg.gravity)) ** 2
    m_fuel = ((B + m_after_cruise**2) * math.tan(theta)) / (
        B**0.5 - m_after_cruise * math.tan(theta)
    )  # math.sqrt(pint.Quantity) is not supported
    return m_fuel.to("kg")

calculate_fuel_consumption_stepclimb_integration ¶

calculate_fuel_consumption_stepclimb_integration(
    m_after_cruise,
    R,
    h,
    M,
    TSFC,
    LD,
    integration_mass_step=100 * ureg.kg,
)

Given a flight distance (=range) \(R\) and basic aircraft performance parameters (see table), returns the fuel mass burned during the flight \(m_f\) [kg] based on a flight schedule where aircraft velocity and altitude are constant during cruise, while allowing for variable thrust-specific fuel consumption and lift-to-drag ratio as a function of Mach number and altitude.

Fuel mass is calculated through numerical integration of the specific air range (SAR) over the flight distance, using the trapezoidal rule:

\[\begin{align*} R &\approx \sum_{i=0}^{n-1} \frac{r_a(m_i) + r_a(m_{i+1})}{2} \cdot (m_{i+1} - m_i) \\ r_a &= \frac{M \cdot a_0 \sqrt{\Theta} \cdot (L/D)}{\text{TSFC} \cdot L} \end{align*}\]

where:

Symbol	Dimension	Description
\(m_{fuel}\)	[mass]	Fuel required for the mission of the aircraft
\(R\)	[distance]	Range of the aircraft (=mission distance)
\(h\)	[length]	Cruise altitude of the aircraft
\(M\)	[dimensionless]	Mach number during cruise
\(TSFC\)	[time/distance]	Thrust Specific Fuel Consumption of the aircraft during cruise
\(L/D\)	[dimensionless]	Lift-to-drag ratio of the aircraft during cruise
\(a_0\)	[speed]	Speed of sound at sea level
\(\Theta\)	[dimensionless]	Temperature ratio at cruise altitude (temperature at cruise altitude divided by temperature at sea level)
\(m_i\)	[mass]	Mass of the aircraft at step \(i\) during cruise, starting with \(m_0 = m_{TO}\) and ending with \(m_n = m_{LDG}\) (= \(m_{after\_cruise}\))
\(g\)	[acceleration]	Gravitational acceleration

Derivation

Equation 19 in Randle et al. (2011) expresses the fuel mass burned \(m_f\) during a flight of distance \(s\) (=range \(R\)) as a function of the takeoff mass \(m_{TO}\). However, in practical applications, the more relevant known parameter is the landing mass after cruise

\[\begin{align} r_a &= -\frac{\text{d}x}{\text{d}m} \\ \text{d}x &= -r_a \text{d}m \\ R = \int_{start}^{end} \text{d}x &= -\int_{m_{TO}}^{m_{LDG}} r_a \text{d}m \\ \end{align}\]

with the specific air range (SAR) defined as:

\[\begin{align} r_a &= -\frac{\text{d}x}{\text{d}m} \\ r_a &= -\frac{\text{d}x/\text{d}t}{\text{d}m/\text{d}t} \\ r_a &= \frac{V}{Q} \\ \end{align}\]

with the definition of true airspeed (=velocity) \(V\) and fuel flow rate \(Q\):

\[\begin{align} V &= M \cdot a_0 \sqrt{\Theta} \\ Q &= \text{TSFC} \cdot D = \text{TSFC} \cdot \frac{L}{(L/D)} \end{align}\]

\[\begin{align} R = \int_{m_{TO}}^{m_{LDG}} \frac{V}{Q} \text{d}m &= \int_{m_{TO}}^{m_{LDG}} \frac{M \cdot a_0 \sqrt{\Theta}}{\text{TSFC} \cdot \frac{L}{(L/D)}} \text{d}m \\ R = \int_{m_{TO}}^{m_{LDG}} \frac{M \cdot a_0 \sqrt{\Theta} \cdot (L/D)}{\text{TSFC} \cdot L} \text{d}m &= \int_{m_{TO}}^{m_{LDG}} \frac{M \cdot a_0 \sqrt{\Theta} \cdot (L/D)}{\text{TSFC} \cdot m \cdot g} \text{d}m \end{align}\]

which can now be solved through numerical integration, using the trapezoidal rule:

\[\begin{equation} R \approx \sum_{i=0}^{n-1} \frac{r_a(m_i) + r_a(m_{i+1})}{2} \cdot (m_{i+1} - m_i) \end{equation}\]

References

Eqn. 13.1aff. and Eqn. 13.28a ff. in Young, T. M. (2018). Performance of the Jet Transport Airplane. John Wiley & Sons. doi:10.1002/9781118534786

Parameters:

Name	Type	Description	Default
`m_after_cruise`	`float`	Mass of the aircraft after cruise segment (eg. OEW + Payload + Crew + Reserves) [mass]	required
`R`	`float`	Range of the aircraft (=mission distance) [distance]	required
`h`	`float`	Cruise altitude of the aircraft [length]	required
`M`	`float`	Mach number during cruise [dimensionless]	required
`TSFC`	`float or Callable`	Thrust Specific Fuel Consumption of the aircraft during cruise, either as a constant value [time/distance] or as a function of Mach number and altitude.	required
`LD`	`float or Callable`	Lift-to-drag ratio of the aircraft during cruise, either as a constant value [dimensionless] or as a function of lift, Mach number and altitude.	required
`integration_mass_step`	`float`	Mass step for numerical integration [mass]. A smaller step will yield a more accurate result, but will also increase computation time. Default is 100 kg.	`100 * kg`

Returns:

Type	Description
`Quantity[mass]`	Required fuel mass [kg]

Raises:

Type	Description
`ValueError`	If the dimensions of the inputs are invalid.
`ValueError`	If the magnitude of the inputs are invalid (eg. negative).

Example

Editor (session: default) Run

import jetfuelburn
from jetfuelburn import ureg
from jetfuelburn.rangeequation import calculate_integrated_range
calculate_integrated_range(
    m_after_cruise=100*ureg.metric_ton,
    R=2000*ureg.nmi,
    h=35000*ureg.feet,
    M=0.78,
    TSFC=17*(ureg.mg/ureg.N/ureg.s),
    LD=18,
)

Output Clear

Source code in jetfuelburn/rangeequation.py

@ureg.check(
    "[mass]",
    "[length]",
    "[length]",
    "[]",
    None,
    None,
    "[mass]",
)
def calculate_fuel_consumption_stepclimb_integration(
    m_after_cruise: pint.Quantity,
    R: pint.Quantity,
    h: pint.Quantity,
    M: float,
    TSFC: pint.Quantity | Callable,
    LD: float | pint.Quantity | Callable,
    integration_mass_step: pint.Quantity = 100 * ureg.kg,
) -> pint.Quantity:
    r"""
    Given a flight distance (=range) $R$ and basic aircraft performance parameters (see table),
    returns the fuel mass burned during the flight $m_f$ [kg] based on a flight schedule 
    where aircraft velocity and altitude are constant during cruise, while allowing for variable thrust-specific fuel consumption
    and lift-to-drag ratio as a function of Mach number and altitude.

    Fuel mass is calculated through numerical integration of the specific air range (SAR) over the flight distance, using the trapezoidal rule:

    \begin{align*}
        R &\approx \sum_{i=0}^{n-1} \frac{r_a(m_i) + r_a(m_{i+1})}{2} \cdot (m_{i+1} - m_i) \\
        r_a &= \frac{M \cdot a_0 \sqrt{\Theta} \cdot (L/D)}{\text{TSFC} \cdot L}
    \end{align*}

    where:

    | Symbol     | Dimension         | Description                                                                                                              |
    |------------|-------------------|--------------------------------------------------------------------------------------------------------------------------|
    | $m_{fuel}$ | [mass]            | Fuel required for the mission of the aircraft                                                                            |
    | $R$        | [distance]        | Range of the aircraft (=mission distance)                                                                                |
    | $h$        | [length]          | Cruise altitude of the aircraft                                                                                          |
    | $M$        | [dimensionless]   | Mach number during cruise                                                                                                |
    | $TSFC$     | [time/distance]   | Thrust Specific Fuel Consumption of the aircraft during cruise                                                           |
    | $L/D$      | [dimensionless]   | Lift-to-drag ratio of the aircraft during cruise                                                                         |
    | $a_0$      | [speed]           | Speed of sound at sea level                                                                                              |
    | $\Theta$   | [dimensionless]   | Temperature ratio at cruise altitude (temperature at cruise altitude divided by temperature at sea level)               |
    | $m_i$      | [mass]            | Mass of the aircraft at step $i$ during cruise, starting with $m_0 = m_{TO}$ and ending with $m_n = m_{LDG}$ (= $m_{after\_cruise}$) |
    | $g$        | [acceleration]    | Gravitational acceleration                                                                                               |


    ??? info "Derivation"

        Equation 19 in Randle et al. (2011) expresses the fuel mass burned $m_f$
        during a flight of distance $s$ (=range $R$) as a function of the takeoff mass $m_{TO}$.
        However, in practical applications, the more relevant known parameter is the landing mass after cruise 

        \begin{align}
        r_a &= -\frac{\text{d}x}{\text{d}m} \\
        \text{d}x &= -r_a \text{d}m \\
        R = \int_{start}^{end} \text{d}x &= -\int_{m_{TO}}^{m_{LDG}} r_a \text{d}m \\
        \end{align}

        with the specific air range (SAR) defined as:

        \begin{align}
        r_a &= -\frac{\text{d}x}{\text{d}m} \\
        r_a &= -\frac{\text{d}x/\text{d}t}{\text{d}m/\text{d}t} \\
        r_a &= \frac{V}{Q} \\
        \end{align}

        with the definition of true airspeed (=velocity) $V$ and fuel flow rate $Q$:

        \begin{align}
        V &= M \cdot a_0 \sqrt{\Theta} \\
        Q &= \text{TSFC} \cdot D  = \text{TSFC} \cdot \frac{L}{(L/D)}
        \end{align}

        \begin{align}
        R = \int_{m_{TO}}^{m_{LDG}} \frac{V}{Q} \text{d}m &= \int_{m_{TO}}^{m_{LDG}} \frac{M \cdot a_0 \sqrt{\Theta}}{\text{TSFC} \cdot \frac{L}{(L/D)}} \text{d}m \\
        R = \int_{m_{TO}}^{m_{LDG}} \frac{M \cdot a_0 \sqrt{\Theta} \cdot (L/D)}{\text{TSFC} \cdot L} \text{d}m &= \int_{m_{TO}}^{m_{LDG}} \frac{M \cdot a_0 \sqrt{\Theta} \cdot (L/D)}{\text{TSFC} \cdot m \cdot g} \text{d}m
        \end{align}

        which can now be solved through numerical integration, using the trapezoidal rule:

        \begin{equation}
        R \approx \sum_{i=0}^{n-1} \frac{r_a(m_i) + r_a(m_{i+1})}{2} \cdot (m_{i+1} - m_i)
        \end{equation}

    See Also
    --------
    [`jetfuelburn.rangeequation.calculate_fuel_consumption_stepclimb_arctan`][]

    References
    ----------
    Eqn. 13.1aff. and Eqn. 13.28a ff. in 
    Young, T. M. (2018). 
    Performance of the Jet Transport Airplane. 
    _John Wiley & Sons_. 
    doi:[10.1002/9781118534786](https://doi.org/10.1002/9781118534786)

    Parameters
    ----------
    m_after_cruise : float
        Mass of the aircraft after cruise segment (eg. OEW + Payload + Crew + Reserves) [mass]
    R : float
        Range of the aircraft (=mission distance) [distance]
    h : float
        Cruise altitude of the aircraft [length]
    M : float
        Mach number during cruise [dimensionless]
    TSFC : float or Callable
        Thrust Specific Fuel Consumption of the aircraft during cruise, either as a constant value [time/distance] or as a function of Mach number and altitude.  
    LD : float or Callable
        Lift-to-drag ratio of the aircraft during cruise, either as a constant value [dimensionless] or as a function of lift, Mach number and altitude.  
    integration_mass_step : float
        Mass step for numerical integration [mass]. A smaller step will yield a more accurate result, but will also increase computation time. Default is 100 kg.

    Returns
    -------
    pint.Quantity [mass]
        Required fuel mass [kg]

    Raises
    ------
    ValueError
        If the dimensions of the inputs are invalid.
    ValueError
        If the magnitude of the inputs are invalid (eg. negative).

    Example
    -------
    ```pyodide install='jetfuelburn'
    import jetfuelburn
    from jetfuelburn import ureg
    from jetfuelburn.rangeequation import calculate_integrated_range
    calculate_integrated_range(
        m_after_cruise=100*ureg.metric_ton,
        R=2000*ureg.nmi,
        h=35000*ureg.feet,
        M=0.78,
        TSFC=17*(ureg.mg/ureg.N/ureg.s),
        LD=18,
    )
    ```
    """
    if integration_mass_step < 1 * ureg.kg:
        raise ValueError("integration_mass_step must be at least 1 kg")
    if m_after_cruise <= 0 * ureg.kg:
        raise ValueError("m_after_cruise must be greater than zero")
    if M <= 0:
        raise ValueError("Mach number must be greater than zero")
    if h < 0 * ureg.meter:
        raise ValueError("Altitude must be non-negative")

    if type(TSFC) == Callable:
        _validate_physics_function_parameters(
            func=TSFC,
            required_params={"M", "h"},
        )
    if type(LD) == Callable:
        _validate_physics_function_parameters(
            func=LD,
            required_params={"L", "M", "h"},
        )
    func_TSFC: Callable = _normalize_physics_function_or_scalar(TSFC)
    func_LD: Callable = _normalize_physics_function_or_scalar(LD)

    m_after_cruise = m_after_cruise.to("kg")
    V = _calculate_airspeed_from_mach(mach_number=M, altitude=h)

    m_current = m_after_cruise
    R_current = 0 * ureg.km

    while R_current < R:
        L_A = m_current * ureg.gravity
        L_B = (m_current + integration_mass_step) * ureg.gravity
        SAR_A = (V * func_LD(L=L_A, M=M, h=h)) / (func_TSFC(M=M, h=h) * L_A)
        SAR_B = (V * func_LD(L=L_B, M=M, h=h)) / (func_TSFC(M=M, h=h) * L_B)
        SAR_avg = (SAR_A + SAR_B) / 2
        delta_R = (SAR_avg * integration_mass_step).to("km")
        R_current += delta_R
        m_current += integration_mass_step

        if R_current >= R:
            R_excess = R_current - R
            m_excess = R_excess / SAR_avg
            m_current -= m_excess
            break

    m_fuel = m_current - m_after_cruise
    return m_fuel.to("kg")