{title2=$e$}

The <span efficiency factor> $e$ compares a wing’s lift-induced drag with a reference induced-drag level at the same lift, span and dynamic <pressure>. In the conventional finite-wing coefficient model,
$$
\boxed{C_{D_i}=\frac{C_L^2}{\pi AR\,e},\qquad
e=\frac{C_L^2}{\pi AR\,C_{D_i}},\quad AR=b^2/S.}
$$
It describes the induced contribution under stated geometry and flow assumptions. NASA’s primary account gives this relation and identifies the elliptic-load ideal in its lifting-line discussion.<Induced Drag Coefficient by NASA Glenn Research Center>[Drag Coefficient and Lifting Line Theory]{ref}

For an ideal planar wing in classical <lifting-line theory>, <elliptic span loading> minimizes induced drag for fixed lift and span, corresponding to $e=1$ in this normalization. Deviations of the loading can increase induced drag and lower $e$. The load distribution depends on twist, section properties and operating conditions as well as planform. An outline alone therefore does not specify a universal efficiency value. Nonplanar systems and different reference conventions require their own interpretation before applying the planar bound.

For an original coefficient example, choose $C_L=0.80$, $AR=8$ and $e=0.80$. Then $C_{D_i}\approx0.031831$. With $q_\infty=500\,\mathrm{Pa}$ and $S=20\,\mathrm{m^2}$, lift is $8000\,\mathrm N$ and induced drag is about $318.3\,\mathrm N$. At $e=1$ with those other quantities fixed, induced drag would be about $254.6\,\mathrm N$. Thus the chosen $e=0.8$ corresponds to a 25-percent induced-drag increase over that reference, not a statement that twenty percent of all input power is lost.

A drag-polar fit illustrates why efficiency definitions can differ. Suppose a model has
$$
C_D=C_{D0}+\left[k_v+\frac1{\pi AR e_i}\right]C_L^2,
$$
where $k_v$ is an additional lift-dependent profile-drag coefficient and $e_i$ is the induced efficiency. Fitting the entire quadratic term as $C_L^2/(\pi AR e_{\rm fit})$ gives
$$
\frac1{e_{\rm fit}}=\frac1{e_i}+\pi AR k_v.
$$
For stipulated $e_i=0.90$, $AR=8$ and $k_v=0.010$, the fitted value is about $0.734$. The algebra does not identify $e_{\rm fit}$ with the induced-only factor. It shows how a total-drag fit can absorb another modeled contribution even when the induced flow itself has not changed.

The inversion is also sensitive to a small extracted drag. At fixed $C_L$ and $AR$, a small fractional error in $C_{D_i}$ gives approximately the opposite fractional error in $e$. If induced drag is obtained by subtracting a large estimated profile contribution from total drag, uncertainty in that subtraction can dominate. Different reference areas or lift conditions also change the comparison.

A proposed test combines measured lift and drag with a separately justified profile-drag estimate or a calibrated wake-based induced-drag inference. Repeat over the intended coefficient range and report the geometry, area convention, flow regime and uncertainty. A single fitted number is useful only alongside the model that separates induced drag from the rest of the measured aerodynamic <force>.


 Span efficiency factor (source code)