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The Three Lift to Drag Ratios

Issue No 46, 2 October 2023

By: Anthony O. Ives

In earlier articles range and endurance [1],[2] have been discussed as well as the various ratios that when they are a maximum give the best range and endurance. There are three ratios that are important for understanding aircraft performance, the first of course is obvious as it is lift to drag ratio the remaining consist of lift coefficient raised to 3/2 over drag coefficient and lift coefficient raised to 1/2 over drag coefficient. The ratios mainly apply to fixed wing aircraft where the ratio of lift coefficient raised to 1/2 over drag coefficient is only really applicable to jet power aircraft as discussed in an earlier article [3]. The maximum ratios are achieved at specific aircraft speeds so its important these speeds can be calculated also.

The first ratio we are going to consider is the lift to drag ratio, all these ratios can be determined from the drag polar equation[4], the drag polar equation is given below:

\[C_D=C_{D0} + k C_L^2 \]

Where is C_D is total drag coefficient, C_D0 is zero lift drag coefficient, k is the drag factor and C_L is the lift coefficient. The drag polar equation can be rearranged and then differentiated with respect to velocity and then the velocity at which the lift to drag ratio is a maximum can be determined. The procedure is used for all the ratios, the maximum lift to drag ratio is given below:

\[\left( \frac{L}{D} \right)_{max} = \left( \frac{C_L}{C_D} \right)_{max} =\sqrt{ \frac{1}{4 C_{D0} k} } \]

Where L is lift and D is drag, the speed required for maximum lift to drag ratio is given below:

\[ V_{(L/D)max} = \left( \frac{2} {\rho} \frac{W}{S} \sqrt {\frac{k}{C_{D0}} } \right)^{\frac{1}{2}} \]

Where ρ is the air density, W is the weight of the aircraft and S is the wing area. The lift to drag ratio is used to determine the maximum range for a propeller driven aircraft [1] and the endurance for jet powered aircraft [3] as dicussed in earlier articles [1],[2] and [3], similarly the maximum ratio of lift coefficient raised to 3/2 over drag coefficient is as below:

\[ \left( \frac{C_L^\frac{3}{2} }{C_D} \right)_{max} = \frac{1}{4} \left( \frac{3}{k C_{D0}^\frac{1}{3}} \right) ^\frac{3}{4} \]

This ratio is mainly used to determine the maximum endurance for propeller driven aircraft [2]. Similarily the speed required for this ratio to be a maximum is:

\[ V_{(C_L^{3/2}/C_D)max} = \left( \frac{2} {\rho} \frac{W}{S} \sqrt {\frac{k}{3 C_{D0}} } \right)^{\frac{1}{2}} \]

\[ V_{(C_L^{3/2}/C_D)max} = \left( \frac{1}{3} \right)^\frac{1}{4} V_{(L/D)max}\]

And again the maximum ratio of lift coefficient raised to 1/2 over drag coefficient is as below:

\[ \left( \frac{C_L^\frac{1}{2} }{C_D} \right)_{max} = \frac{3}{4} \left( \frac{1}{3 k C_{D0}^3} \right) ^\frac{1}{4} \]

The ratio is used to determine the maximum endurance for jet powered aircraft [3]. And again the speed required for this ratio to be a maximum is:

\[ V_{(C_L^{1/2}/C_D)max} = \left( \frac{2} {\rho} \frac{W}{S} \sqrt {\frac{3 k}{C_{D0}} } \right)^{\frac{1}{2}} \]

\[ V_{(C_L^{1/2}/C_D)max} = 3^\frac{1}{4} V_{(L/D)max}\]

Using the information from an earlier articles [4] and [5] we can now calculate the three maximum ratios and the airspeed required to make these ratios a maximum. They are given in the table below along with the information required to calculate them:

Symbol	Property	Example Value	Units
k	Drag Factor	0.0531	none
C_D0	Zero Lift Drag Coefficient	0.06	none
ρ	Air Density	1.2256	kgm^-3
W	Aircraft Weight	2.07	kg
S	Aircraft Wing Area	0.736	m³
\(\left( \frac{L}{D} \right)_{max}\)	Lift to Drag Ratio	8.858	None
\(V_{(L/D)max}\)	Airspeed required for Maximum Ratio	6.508	ms^-1
\(\left( \frac{C_L^\frac{3}{2} }{C_D} \right)_{max}\)	Coefficient Lift raised to 3/2 to Drag Ratio	13.699	None
\(V_{(C_L^{3/2}/C_D)max}\)	Airspeed required for Maximum Ratio	4.945	ms^-1
\(\left( \frac{C_L^\frac{1}{2} }{C_D} \right)_{max}\)	Coefficient Lift raised to 1/2 to Drag Ratio	9.793	None
\(V_{(C_L^{1/2}/C_D)max}\)	Airspeed required for Maximum Ratio	8.565	ms^-1

The stall speed was calculated at 6.71 ms^-1 in the previous article [5] which means in this case that two of the ratios can not be practically achieved due to the required speed being below the stall speed. This can happen based on the design of the aircraft, so my choice of aspect ratio, wing loading, etc is probably not that good. The air density hence altitude the aircraft is operating can also have an impact on the airspeed required to give you maximum ratios.

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References:

[1] http://www.eiteog.com/EiteogBLOG/No6EiteogBlogRange.html

[2] http://www.eiteog.com/EiteogBLOG/No7EiteogBlogEndurance.html

[3] http://www.eiteog.com/EiteogBLOG/No41EiteogBlogJet.html

[4] http://www.eiteog.com/EiteogBLOG/No2EiteogBlogDragCD.html

[5] http://www.eiteog.com/EiteogBLOG/No1EiteogBlogLiftCL.html

[6] Aircraft Performance & Design, John D. Anderson Jr., 1999, McGraw Hill

[7] Civil Jet Aircraft Design, Lloyd R. Jenkinson, Paul Simpkin, Darren Rhodes, 1999, Butterworth Heinemann

[8] Aircraft Performance (Cambridge Aerospace Series, Series Number 5), W. Austyn Mair, David L. Birdsall, 1996, Cambridge University Press

Disclaimer: Eiteog makes every effort to provide information which is as accurate as possible. Eiteog will not be responsible for any liability, loss or risk incurred as a result of the use and application of information on its website or in its products. None of the information on Eiteog's website or in its products supersedes any information contained in documents or procedures issued by relevant aviation authorities, manufacturers, flight schools or the operators of aircraft, UAVs.