Mathematics For Engine By R. G. Huggins 4915 S. Detroit Tulsa. Okla. 74105

T

O BE SUCCESSFUL when experimenting with engines for amateur-built aircraft, it is essential that one know how to go about making computations of various kinds. For example, Volkswagen and other European engines are built to the metric system of measurement and unless one is adept at converting metric dimensions to decimal equivalents it is hard to compare such an engine with those built in this country. Bore and stroke measurements in European engines are stated in millimeters and displacement in cubic centimeters, one cubic centimeter equalling 1000 cubic millimeters. The formula for converting bore and stroke measurements from millimenters to inches is: MM

——— = inches 25.4 To illustrate, a cylinder bore of 77 millimeters has a diameter of 77 ———— = 3.03 inches 25.4 Inches are converted to millimeters by the formula: 25.4 x inches = millimeters To illustrate, the Corvair engine's piston stroke of 2.60 in. would be 25.4 x 2.60 = 66 mm. As one cubic inch equals 16.387 cubic centimeters, the formula for converting engine displacement expressed in cubic centimeters to cubic inches is: cc's ———— — cubic inches. 16.387 For example, consider the VW engine of 1192 cc. displacement: 1192 ———— -. 72.74 cubic inches. 16.387 To convert cubic inches to cubic centimeters the formula is: Cubic inches x 16.387 = cc's. A 145 cubic inch Corvair engine has: 145 x 16.387 -- 2376 cubic inches displacement. An experimenter should be able to compute the displacement, or volume swept by all of the pistons, of his engine. As one of the most common ways of increasing power is by increasing the displacement, it is necessary to know how to compute displacement so that changes in an engine resulting from increasing the bore, or the stroke, or both, can be determined. Factors involved in calculating displacement are the bore (diameter of the cylinders), the length of the piston stroke, the number of cylinders in the engine and a constant, .7854, which is one-fourth of 3.1416 or Pi. Pi is used in formulas dealing with the dimensions of circles. Actually, the engine displacement formula is the same as the standard one for computing the volume of a cylinder, the result being multiplied by the number of cylinders. The cross-sectional area of a cylinder is determined 8

JUNE 1969

and then the volume of one cylinder is found by multiplying the area by the stroke length, which is the equivalent of the length of the cylinders. Multiplying by the number of cylinders gives the engine's total displacement. The formula is: Bore Diameter x Bore Diameter x .7854 x Stroke Length x Number of Cylinders = Displacement. This formula works with either inches or centimeters. One inch equals 2.54 centimeters; one cubic inch equals 16.38 cubic centimeters. Consider a standard VW engine of 1192 cubic centimeters displacement. One centimeter equals ten millimeters, so we can point off one place and the formula becomes: 7.7 x 7.7 x .7854 x 6.4 x 4 = displacement in cc. 59.29 x .7854 = 46.566 46.566 x 6.4 = 298.024 298.024 x 4 = 1192 cubic centimeters displacement.

To convert to cubic inches we divide the cc. by 16.38 or 1192/16.38 = 72.74 cubic inches. It is possible to install Corvair cylinders on the VW engine, with suitable small changes and procedures. The bore will become 3-7/16 in. and the stroke will remain the same 2.520 in. The formula with the bore measurement

in fractions converted to decimals becomes: 3.4375 x 3.4375 x 2.52 x 4 = 93.548 cu. in. displacement. An engine's compression ratio is computed by comparing the cylinder's volume, which we call its displacement in this kind of talk, with the total volume of the cylinder and its combustion chamber. Cylinder volume can be determined mathematically, but because of the typical combustion chamber's highly irregular shape, it is usually preferred to measure the volume by filling it with a suitable liquid poured from a graduate. Combustion chamber volume can be determined with the cylinder head in place or removed from the engine. An accompanying sketch shows how it is done with the head in place, such as when checking racing machines before or after a race for compliance with race rules. To do it with a head that has been removed from the engine, install valves and the sparkplug and support the head so that its gasket surface is truly level. Then fill a graduate with water, light oil or kerosene and pour the liquid into the combustion chamber. Do not overfill the chamber, something that is quite easy to do when using water or other liquid having high surface tension — the water level will rise appreciably above the gasket surface of the head before finally spilling over onto the gasket surface. If using a liquid with high seepage properties such as kerosene, make sure the valves are not leaking fluid during the pouring and measuring process. The volume of the combustion chamber is found by subtracting the quantity of fluid left in the graduate after filling the combustion chamber from the amount it originally held. Most graduates are marked in cubic centimeters, which is convenient when working with VW's but making it necessary to convert the results to inches when

measuring domestic engines. So long as all volumes used in the figuring are either metric or inches, the compression ratio arrived at is the same either way. If the piston has a flat top, there's no problem. If it has a domed top. the volume of the dome must be taken into account. Similarly, it is necessary to remember the thickness of the cylinder head gasket. When measuring for compression ratio with the cylinder head installed on the engine, the method shown in the sketch automatically allows for domed pistons and the head gasket. When

working with the head off, compute the volume of the cylinder opening in the gasket by multiplying its area by its thickness and add the resulting volume to the combus-

tion chamber volume. When working with two-cycle en-

gines, it is usual to compute compression ratio by using the volume of the cylinder remaining after all cylinder wall ports have been covered by the piston on its up-stroke. Sometimes it is found that an engine is built in such a way that the pistons do not travel completely up the cylinder bore, or that their tops go a little above the cylinder's gasket face when at top dead center. Rotate your engine and observe piston position when it reaches top dead center; if it stops a fraction of an inch below or above the cylinder-to-head parting line, compute the minus or plus volume by measuring the distance from the parting line to the piston top with a depth micrometer or similar device and figure as follows:

Bore x Bore x .7854 x Distance from top of piston to top of cylinder = Volume. This volume is added to the volume of the combustion chamber and head gasket opening, to get final combustion chamber volume. To obtain compression ratio: Final combustion chamber volume / Cylinder volume Final combustion chamber volume = Compression Ratio. For an example, consider a VW engine fitted with Corvair cylinders. Assume that the final combustion chamber volume is 3.34 cubic inches and the cylinder volume is 23.38 cu. in. Applying these figures to the formula, we get the equation: 3.34 + 23.38

chamber from a graduate to determine its volume. Volume

of cylinder can also be obtained by this method, by positioning piston at bottom dead

center, adding more fluid and subtracting amount in combustion chamber to find amount in cylinder.

or 33,000 pounds lifted one foot in one minute both require the expenditure of one horsepower. Because the force exerted by an engine crankshaft is done in a circular direction rather than in a straight line, the constant "2 pi" is inserted in the formula. This makes the formula for horsepower developed by an engine: 2 pi x torque x rpm

————————————— = Horsepower

33,000 By eliminating the 2 pi constant and reducing the 33,000 accordingly, the formula is simplified to: Torque x rpm ———————— = Horsepower 5252

As an example, let's compute the horsepower output of a 1961 VW engine at its advertised maximum torque output of 64 foot pounds at 2400 rpm. Applying these figures to the formula we get the equation:

____________

___

3.34

It is wise to check out figures obtained by the head-off method by doing it again after the head had been installed on the engine, with the method shown in the illustration. Horsepower and torque are commonly used measures of engine performance. They are related to the extent that one cannot exist without the other. Torque is a measure of the amount of work an engine can do and horsepower is the measure of the amount of work done in a given time. The time factor for horsepower computations is the crankshaft's speed as measured in revolutions per minute. Measurement of an engine's torque requires the use of a device called a dynamometer. Such an item is a large and expensive thing and is something that only engineering laboratories can afford. Even though the average experimenter can't measure his engine's torque output, he should understand the formulas that apply to torque and horsepower. Torque is measured in foot pounds. A foot pound is the force exerted by one pound acting on a lever one foot long. 33,000 foot pounds of work done in one minute

equals one horsepower, and horsepower can be developed

153,000

5252

5252

For computing torque when horsepower and rpm's are known, the formula is: Horsepower x 5252 ———————————— = Torque RPM To compute the torque delivered by a VW when it is delivering its maximum advertised horsepower of 40 at 4000 rpm., the formula is: 40 x 5252

210,080

4000

400

——————— = ———— = 52.5 foot pounds

Q

The compression ratio is 8 to 1.

64 x 2400

——————— = ———— = 29 Horsepower

26.72 ___

3.34

With piston at top dead center, fluid is poured into combustion

of torque.

"GULFHAWK II" COMING TO EAA AIR MUSEUM

Certainly one of America's most famous airplanes is the Grumman G-22 "Gulfhawk II" which flew many shows during the years preceding and immediately following World War II. Guided by the skillful hands of the legendary late Alford Williams, the tubby biplane even saw service during the war when it was used to demonstrate precision flying to the military flying cadets. The "Gulfhawk II", sponsored by Gulf Oil Corp., was presented to the Smithsonian Institution's National Air Museum, whereupon it was placed in temporary storage awaiting suitable display. This has not yet presented itself, and the airplane has been loaned to the EAA Air Museum through the courtesy of National Air and Space Museum Director S. Paul Johnston where the space is available and it can be readied for display with a minimum of preparatory effort. The story of the "Gulfhawk II" appeared in the January, 1969 SPORT AVIATION.

by any means. One pound lifted 33,000 feet in one minute

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9