Is there a parameter that links the areas of the fuselage and control surfaces? [closed]

Question

Do aircraft designers look at the ratio of how much fuselage surface area can be loaded onto a given unit of wing and tail surface area including elevator, aileron, rudder etc? (Possibly, the surfaces of propellers and internal parts of jets, may be excluded while making such calculations.)

Is there such a parameter, and if so, what are the normal values?

This question is for normal fixed-wing and rotary-wing aircraft, and mostly commercial aircraft.

This question is NOT about fighter jets, space-shuttles, canard-wing designs, flying wings, lifting body and other cutting-edge-technology stuff.

Answer 1

I do not know a method that relates the fuselage area to empennage sizes, but there is rule that can be used to estimate the size of the vertical and horizontal tail. It is using the so called tail volume coefficient. For the horizontal tail given by:

$$ V_h = \frac{S_h \cdot l_h}{S c} $$

With $V_h$, the horizontal tail volume coefficient, $S_h$ the horizontal tail surface, $l_h$ the horizontal distance between the c.o.g. and the tail, $S$ the wing area and $c$ the average wing chord. Basically, this coefficient says that the moment induced by the wing scales with $S$ and $c$, whereas the counteracting moment given by the tail scales with the area of the horizontal wing multiplied by the distance to the c.o.g., and that there should a certain ratio between the two.

A well behaved aircraft typically has a $V_h$ which falls in the following range $^{[1]}$

$$V_h = 0.30 ... 0.60$$

If $V_h$ is too small, the aircraft’s pitch behavior will be very sensitive to the CG location. It will also show poor tendency to resist gusts or other upsets, and generally “wander” in pitch attitude, making precise pitch control difficult. $^{[1]}$

We can check the values for some commercial aircraft here (I chose Airbus, but it should hold for others as well) :

$$\begin{array}{|c|c|} \hline \text{Model} & \text{$V_h$} \\ \hline \text{A330-200} & 0.957 \\ \hline \text{A330-300} & 0.791 \\ \hline \text{A340-200} & 0.733 \\ \hline \text{A340-300} & 0.791 \\ \hline \text{A340-500} & 0.729 \\ \hline \text{A340-600} & 0.729 \\ \hline \end{array}$$

You can see that the values are already higher than given by the guideline, showing the issue of the generalization. Another fact that causes the deviations, is the fact that the Airbus aircraft are designed in families. Thus, they design an aircraft, and then add some extra fuselage length to generate a larger family member. This will possibly drive the tail volume coefficient away from the ideal case.

For the vertical tail there is an equivalent form:

$$V_v = \frac{S_v l_v}{S b}$$

With $V_v$, the horizontal tail volume coefficient, $S_v$ the horizontal tail surface, $l_v$ the vertical distance between the c.o.g. and the tail, $S$ the wing area and $b$ the average wing span.

A well behaved aircraft typically has a $V_v$ which falls in the following range $^{[1]}$ $$V_v = 0.02 ... 0.05$$

If $V_v$ is too small, the aircraft will tend to oscillate or “wallow” in yaw as the pilot gives rudder or aileron inputs. This oscillation, shown in Figure 5, is called Dutch Roll, and makes precise directional control difficult. A Vv which is too small will also give poor rudder roll authority in an aircraft which uses only the rudder to turn. $^{[1]}$

Again checking the values:

$$\begin{array}{|c|c|} \hline \text{Model} & \text{$V_v$} \\ \hline \text{A330-200} & 0.057 \\ \hline \text{A330-300} & 0.059 \\ \hline \text{A340-200} & 0.055 \\ \hline \text{A340-300} & 0.059 \\ \hline \text{A340-500} & 0.049 \\ \hline \text{A340-600} & 0.049 \\ \hline \end{array}$$

Here the values are quite close to the general guideline.

Another interesting plot is shown by This tail design page by Stanford

Here it shows that the tail volume correlates quite nicely to fuselage height and length.

Image showing correlation of aircraft vertical tail volume as a function of fuselage maximum height and length.

It should be noted that aircraft aren't really designed by these guidelines anymore, all the control surfaces are designed such that they provide the control authority necessary, not to match a general rule.

$^{[1]}$: Lab 8 Notes – Basic Aircraft Design Rules Link

Answer 2

The engineer strives to give the aircraft a positive $c_{n_{\beta}}$ by adjusting the size of the vertical tail. A good value is $c_{n_{\beta}} = 0.1$ when the reference length is the wingspan, or $c_{n_{\beta}} = 0.2$ when the half span is used as reference length.

This is just a fancy way of saying that the aircraft should have natural weathervane stability and correct sideslip angles by turning into the wind. A fuselage by itself is mostly unstable, because the aerodynamic center of slender bodies is located close to their tip. Any side force on the fuselage will thus act ahead of the center of gravity. A compensating force acting behind the center of gravity is needed. Since not the forces themselves, but rather their moments around the center of gravity count, you cannot simply compare areas, but the product of area, lever arm and lift curve slope, which in turn is determined by the aspect ratio of the area concerned.

The higher aspect ratio of a vertical tail surface (higher at least when compared to the fuselage) will allow to make the tail smaller, but causes an earlier stall of the vertical tail. As a consequence, the stabilizing side force on the tail stops to increase at a sideslip angle where the destabilizing side force on the fuselage will still blithely increase with sideslip angle. To improve stability at high sideslip angles, strakes are added ahead of the vertical tail. Their surface will do little at small sideslip angles, but be very important at higher angles.

As you can see, comparing areas is not enough.