1.2 Ratios, Proportions, and Percentages

Ratios and Proportions in Aviation

In aviation, ratios and proportions are used to describe comparative relationships between physical quantities. A ratio is a comparison of two quantities, representing how many times one value contains another. Ratios can be expressed as a fraction (a/b) or with a colon (a:b). A proportion is a statement that two ratios are equal (a/b = c/d).

Wing Aspect Ratio

In aerodynamics, the aspect ratio (AR) of an aircraft wing is a critical design factor that compares the wing's span (tip-to-tip length) to its chord (width). For a rectangular wing, the formula is:

AR = Wing Span / Wing Chord

For non-rectangular wings, the formula is modified to use the wing area (S) and span (b):

AR = (b squared) / S

A high aspect ratio wing is long and narrow. Examples include sailplanes (gliders) and high-altitude aircraft like the U-2. These wings generate high lift and low induced drag, making them extremely efficient at subsonic speeds. Conversely, a low aspect ratio wing is short and stubby, typical of jet fighters. Low aspect ratio wings are less aerodynamically efficient at low speeds but offer superior maneuverability and structural strength for supersonic flight.

Piston Engine Compression Ratio

In reciprocating aircraft engines, the compression ratio (CR) is a comparison of the cylinder volume when the piston is at the bottom of its stroke (Bottom Dead Center, or BDC) to the cylinder volume when the piston is at the top of its stroke (Top Dead Center, or TDC). The volume at BDC is the sum of the displacement volume (Vd, the volume swept by the piston) and the clearance volume (Vc, the remaining space in the cylinder head). The volume at TDC is simply the clearance volume (Vc).

CR = (Vd + Vc) / Vc

For example, if a cylinder has a displacement volume of 80 cubic inches and a clearance volume of 10 cubic inches, its compression ratio is:

CR = (80 + 10) / 10 = 90 / 10 = 9.0:1

A higher compression ratio extracts more mechanical energy from the fuel-air mixture, improving thermal efficiency. However, high compression ratios generate extreme heat and pressure, which can cause the fuel-air mixture to detonate (explode prematurely) rather than burn smoothly. Piston engines with high compression ratios require high-octane fuels, such as Avgas 100LL, which are formulated to resist detonation.

Propeller Gear Ratios

Aircraft engines achieve maximum horsepower at high rotational speeds (RPM). However, if a propeller is directly connected to the crankshaft, high engine RPM will cause the propeller blade tips to exceed the speed of sound. Supersonic propeller tips create massive drag, noise, and structural stress while losing aerodynamic efficiency. To resolve this, engines use a propeller reduction gearbox.

The relationship between engine crankshaft speed and propeller speed is governed by the gearbox's gear ratio. The gear ratio is determined by the number of teeth on the driving gear (connected to the crankshaft) and the driven gear (connected to the propeller shaft):

Gear Ratio = Teeth of Driven Gear / Teeth of Driving Gear

If the driving gear has 48 teeth and the driven gear has 72 teeth, the ratio is 72:48, which simplifies to 3:2 (or 1.5:1). This means the engine crankshaft must rotate 1.5 times for every 1 rotation of the propeller. The rotational speed is inversely proportional to the gear ratio:

Propeller RPM = Engine RPM / (Driven Gear Teeth / Driving Gear Teeth)

If the engine operates at 2,700 RPM, the propeller speed is:

Propeller RPM = 2700 x (48 / 72) = 1800 RPM

Fuel-Mix Ratios

The fuel-mix ratio defines the proportion of air and fuel by mass delivered to the engine cylinders. The ideal ratio for complete combustion is the stoichiometric ratio, which is 14.7:1 (14.7 pounds of air to 1 pound of fuel).

• Rich Mixture: If the ratio decreases (e.g., 12:1), the mixture has excess fuel. Rich mixtures are used during takeoff and climb because the unburned fuel helps cool the engine cylinders and prevents detonation.
• Lean Mixture: If the ratio increases (e.g., 16:1), the mixture has excess air. Lean mixtures are used during cruise flight to maximize fuel economy, though they run hotter.

Proportions: Direct and Inverse

Understanding the difference between direct and inverse proportions is essential for analyzing physical aircraft behaviors:

• Direct Proportion: Two quantities change in the same direction. If one increases, the other increases proportionally. For example, aerodynamic lift is directly proportional to air density. If air density decreases (as altitude increases), lift decreases proportionally (assuming velocity and wing area remain constant).
• Inverse Proportion: Two quantities change in opposite directions. If one increases, the other decreases. The classic aviation example is Bernoulli's principle in a Venturi tube: fluid velocity is inversely proportional to fluid pressure. As air enters the narrow throat of a Venturi, its velocity increases, and its static pressure drops.

Averages and Percentages

Averages (Arithmetic Mean)

Technicians calculate averages to monitor system performance. The arithmetic mean is found by summing a set of values and dividing by the total count.

During engine testing, a technician monitors Cylinder Head Temperatures (CHTs) across all cylinders. If a six-cylinder engine records CHTs of 380, 395, 385, 410, 390, and 400 degrees Fahrenheit, the average CHT is:

Average = (380 + 395 + 385 + 410 + 390 + 400) / 6 = 2360 / 6 = 393.3 degrees Fahrenheit

An individual cylinder running significantly hotter or colder than this average indicates a localized problem, such as a clogged fuel injector or an air leak.

Percentages and Tolerances

Percentages represent fractions out of 100. In maintenance, percentages are critical for expressing structural and component wear tolerances. Aircraft parts have nominal dimensions when new, and manuals dictate the maximum allowable wear as a percentage of the original dimension.

For example, a steel brake disc has a nominal thickness of 0.250 inches. The manufacturer specifies a maximum wear limit of 12%. This means the disc can lose up to 12% of its thickness before it must be discarded. To find the minimum allowable thickness:

1. Calculate the wear limit value: 12% of 0.250 = 0.12 x 0.250 = 0.030 inches.
2. Subtract the wear limit from the nominal thickness: 0.250 - 0.030 = 0.220 inches.

Alternatively, calculate the remaining thickness directly by multiplying the nominal value by the remaining percentage (100% - 12% = 88%):

0.250 x 0.88 = 0.220 inches.