# 7.1.9: Range and endurance

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In this section, we study the range and endurance for an aircraft flying a steady, linear-horizontal flight.

• The range is defined as the maximum distance the aircraft can fly given a quantity of fuel.
• The endurance is defined as the maximum time the aircraft can be flying given a quantity of fuel.

• Consider a symmetric flight in the horizontal plane.
• $$\chi$$ can be considered constant.
• The aircraft performs a leveled wing flight, i.e., $$\mu = 0$$.
• There is no wind.
• The velocity of the aircraft is constant.

The 3DOF equations governing the motion of the airplane are:

$T = D,\label{eq7.1.9.1}$

$L = mg,\label{eq7.1.9.2}$

$\dot{x}_e = V,\label{eq7.1.9.3}$

$\dot{m} = -\eta T.\label{eq7.1.9.4}$

Equation ($$\ref{eq7.1.9.4}$$) means that the aircraft losses weight as the fuel is burt, where $$\eta$$ is the specific fuel consumption. Notice that Equation ($$\ref{eq7.1.9.4}$$) is just valid for jets.

The specific fuel consumption is defined in different ways depending of the type of engines:

• jets: $$\eta_j = \tfrac{-dm/dt}{T}$$.
• Propellers: $$\eta_p = \tfrac{-dm/dt}{P_m} = \tfrac{-dm/dt}{TV}$$, where $$P_m$$ is the mechanical power.

Focusing on jet engines, operating with Equation ($$\ref{eq7.1.9.1}$$, $$\ref{eq7.1.9.2}$$, $$\ref{eq7.1.9.3}$$, $$\ref{eq7.1.9.4}$$), considering $$E = L/D$$, and taking into account the initial state $$(\cdot)_i$$, and the final state $$(\cdot)_f$$ we obtain the distance and time flown as:

$x_e = -\int_{m_i}^{m_f} \dfrac{V}{\eta_j T} dm = -\int_{m_i}^{m_f} \dfrac{1}{\eta_j g} VE \dfrac{dm}{m},$

$t = -\int_{m_i}^{m_f} \dfrac{1}{\eta_j T} dm = -\int_{m_i}^{m_f} \dfrac{1}{\eta_j g} E \dfrac{dm}{m},$

In order to integrate such equations we need to make additional assumptions, such for instance consider constant specific fuel consumption and constant aerodynamic efficiency (remember that the velocity has been already assumed to be constant).

Range and endurance (maximum distance and time, respectively) are obtained assuming the aircraft flies with the maximum aerodynamic efficiency (given the weights of the aircraft and given also that for a weight there exists an optimal speed):

$x_{e\ \max} = \dfrac{1}{\eta_j g} VE_{\max} \ln \dfrac{m_i}{m_f},$

$t_{\max} = \dfrac{1}{\eta_j g} E_{\max} \ln \dfrac{m_i}{m_f},$

7.1.9: Range and endurance is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Manuel Soler Arnedo via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.