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DOWNLOADIssue 7, 19 September 2022
By: Anthony O. Ives
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Endurance is the maximum time an aircraft can fly for. For most aircraft endurance is important as an aircraft may have to wait to land and need to loiter until a landing slot becomes available. However, some aircraft their specific mission may require them to loiter over a certain location such as surveillance aircraft. For propeller driven aircraft, drag power ratio has the most influence over endurance. Drag power is the amount of engine power required to overcome the aircraft drag.
Similar to as we did for our discussion for range will we look at battery powered aircraft first and in more detail. The endurance equation for battery powered aircraft is as follows:
\[E={3.6 C_b V_{req} \over P_{req} } \]
Where E is endurance which the maximum time the aircraft can fly. Preq is the required power to overcome the drag in cruise. Vreq is the battery voltage required as defined in the article on range, see Ref [1].
Battery capacity required for a defined endurance is given by the following equation:
\[C_b={E P_{req} \over 3.6 V_{req} } \]
As we did for range, Ref[1] we do an example calculate we have included the range calculation from Ref[1] assuming the aircraft has to travel that distance and then need to loiter. The following table shows how to use the endurance equation for battery powered aircraft to calculate the required battery capacity using some assumed and calculated values from previous articles.
| Symbol | Property | Example Value | Units |
|---|---|---|---|
| R | Aircraft range | 28x1000=28000 | m |
| E | Aircraft endurance | 30x60=1800 | s |
| VReq | Battery voltage required | 11.1 | N |
| TReq | Thrust required | 6.292 | N |
| P | Propulsive power required | 134 | W |
| Cb | Battery capacity required for Range | 1800*134/(3.6*11.1)=6036 | mAhr |
| Cb | Battery capacity required for Endurance | 28000*6.292/(3.6*11.1)=4408.8 | mAhr |
| Cb | Total battery capacity required | 4408.8+6036.0=10444.8 | mAhr |
Similar to range in Ref [1], in most cases when you are designing an aircraft you may not know the power required intially, you may just know a typical lift to drag ratio for your type of aircraft and possibility a design speed. The endurance for battery powered can therefore be more useful in general form as below:
\[E={\frac{w_f}{w_0} \frac{1}{V} \frac{L}{D} 3.6 C_{sb}}\]
Where L/D is the lift to drag ratio. V is aircraft velocity. Csb is the specific battery capacity. wf/w0 is the ratio of fuel mass to total aircraft mass, in the case of battery powered aircraft the fuel mass is the battery mass. This equation is also discussed in different form to determine the fuel mass ratio in Ref [2]. Specific battery capacity is also discussed in Ref [2].
Similar to range for an propeller driven aircraft with expendable fuel such as gasoline or jet fuel the equation is similar but in a slightly different form:
\[E={\frac{1}{V} \frac{L}{D} \frac{\eta} {c_{fuel}} log_e \frac {1}{1-\frac{w_f}{w_0}}}\]
As explained in Ref [1] loge is natural logarithm which you can work out on a scientific calculator or a computer such microsoft excel. cfuel is specific fuel consumption and η is propulsion efficiency.
Endurance is slightly more complicated than range due to fact it depends on drag power for propeller aircraft. Drag power is related to a very unusual ratio CL3/2/CD. The endurance equations presented up to this point is assuming constant velocity. If we assume a constant air density therefore the endurance equation can take on a different but complicated form as below:
\[E={\frac{\eta}{c_{fuel}} \sqrt{\frac{2 \rho S}{w_0}} \frac{C_L^{\frac{3}{2}}}{C_D} \sqrt{\frac{\frac{w_f}{w_0}}{1-\frac{w_f}{w_0}}}}\]
Which can also take the following in terms of drag power ratio directly:
\[E={2 \frac{\eta}{c_{fuel}} \frac{L}{DV} \sqrt{\frac{\frac{w_f}{w_0}}{1-\frac{w_f}{w_0}}}} \]
Where ρ is air density and S is wing area. The relevant of this equation will be explained in more detail in future artical discussing all the various lift coefficent to drag coefficent ratios.
Endurance for an propeller aircraft driven is higher with a low DV/L ratio which is also known as the drag power ratio. Similar to range lower fuel burn helps along with a high propulsion efficiency. The equations for endurance apply to helicopters or rotary aircraft in same way as range. The endurance equations in the same way most aerodynamic equations do, should help summarise the main factors influencing endurance. Ref [3],[4] and [5] discuss endurance in more detail.
Please leave a comment on my facebook page or via email and let me know how if you understand what factors influence aircraft endurance and if you understand what drag power is.
References:
[1] http://www.eiteog.com/No6EiteogBlogRange.html
[2] http://www.eiteog.com/No1EiteogBlogLiftCL.html
[3] Aircraft Performance & Design, John D. Anderson Jr., 1999, McGraw Hill
[4] Civil Jet Aircraft Design, Lloyd R. Jenkinson, Paul Simpkin, Darren Rhodes, 1999, Butterworth Heinemann
[5] Aircraft Performance (Cambridge Aerospace Series, Series Number 5), W. Austyn Mair, David L. Birdsall, 1996, Cambridge University Press
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