THE ART OF ESTIMATING AND SCALING

Sometimes one must give a reasonable estimate of an airplane’s charac­teristics or performance without having all the facts at hand. It is therefore a good idea to commit to memory a few principles and numbers. The “square-cube” scaling law is a good one to remember. For two geometrically similar airplanes designed to the same stress levels and using the same materials, one would expect their areas to be proportional to the characteristic

length squared and their volumes, and hence weight, proportional to the length cubed.

Socl2

Woe Iі

It follows that / a wm. One would therefore expect the wing loading of aircraft to vary as

w

~ « W113 (7.65)

Figure 7.33 was prepared with Equation 7.65 in mind. This figure in­dicates that while the square-cube law can be helpful in estimating the gross weight of an airplane, other factors must also be considered. For performance reasons, wing loadings are sometimes made purposefully higher or lower than the average. Generally, the aircraft with higher cruising speeds lie on the high side of the shaded portion in Figure 7.33. This upper boundary is given by

– 2.94 (Wm – 6) psf

= 85.5 Wm – 9.9) N/m2 (7.66)

Figure 7.33 Square-cube law.

The lower boundary is approximated by,

1.54(W1,3-6)psf

= 44.8 (W1,3 – 9.9) N/m2 (7.67)

The foregoing must be qualified somewhat. Scaling, such as this, is valid only if pertinent factors other than size remain constant, for example, the struc­tural efficiency of materials. Also, for purposes of their mission, aircraft are designed for different load factors.

In sizing an aircraft, it is also of value to note that the empty weights of aircraft average close to 50 or 60% of the design gross weights as shown in Figure 7.34. Thus, knowing the payload and fuel and having some idea of the aerodynamic “cleanliness” of the aircraft, one can undertake a preliminary estimate of its weight and performance.

For example, suppose we are designing a four-place, light aircraft with fixed gear. Let us arbitrarily decide on 300 lb of fuel. The gross weight will be

Gross weight (1VG) X 10 3 lb Figure 7.34 Relationship between gross and empty weight.

474 AIRPLANE PERFORMANCE approximately

Gross weight = Weight empty + Payload + Fuel weight

W = WE + WPL + WF (7.68)

Assuming each passenger and baggage to weigh 200 lb gives

W= WE + 800 + 300

But WE — 0.55 W, so that

W = 2400 lb

From Figure 7.34 for this weight, the wing loading should be ap­proximately

WIS = 14.7 psf

This results in a wing area of S = 163 ft2. If we decide on a low-wing airplane,

Table 7.6 Summary of Approximate Relationships

Note. Output from turbojet engines, approximately proportional to density ratio, o. For piston engines, P — P0 (o–0.1)/0.9.

then an e of 0.6 is reasonable. Also, for a fixed-gear aircraft, a parasite CD of 0.037 was recommended in Chapter Four. We are now in a position to construct power-required curves for various altitudes. The next step would be to select a power plant so that performance estimates can be made. In so doing, we would make use of a typical BSFC of 0.5 lb/bhp-hr for piston engines (unless performance curves on the particular engine selected were available). The value of developing a “feeling” for reasonable values for airplane parameters should be obvious. Table 7.6 summarizes some of these.