# Engineering of Water Systems

Published On: June 21, 2024Categories: Pumps and Water Systems, The Water Works

### Engines: Displacement, Horsepower, Fuel Consumption, and More.

By Ed Butts, PE, CPI

We introduced in April the concept of internal combustion engines and their function. As a continuation on the topic, we will review this month the definitions of displacement (bore and stroke), cylinder pressure, engine horsepower and torque relationships, and fuel consumption.

### Displacement and Ratios

Figure 1. Cylinder and piston dimensions.

Although engine design varies between manufacturers, certain principles of engine design and performance are reasonably constant and uniform for all.

One is based on the horsepower output from the volume of the total displacement contained within the engine. The engine’s displacement is defined as the total interior swept volume (area of the piston × the length of stroke) occupied by the total number of cylinders. Displacement is often expressed in metric units as cubic centimeters (CCs) or in imperial units as cubic inches (CI, and more appropriately referred to as the cubic inch displacement, or CID). However, in recent years, the term liters has been commonly used in recognition of the established use of the metric system of universal measurements with SI units (i.e., International System of Units) being used.

For estimating or design purposes, the displacement of a single cylinder is often calculated as the base power output and then multiplied by the total number of cylinders contemplated for the engine to obtain the total horsepower of the engine.

The two components of displacement are the bore and the stroke (Figure 1). An engine’s bore is the diameter of each cylinder while the stroke is the distance within the cylinder that the piston travels. The bore size directly affects the engine’s capacity to intake and exhaust gases, as well as the overall power output. The stroke length determines the engine’s displacement, which is the total volume of air and fuel mixture the engine can intake and exhaust during one complete cycle.

Figure 2. Engine bore/stroke ratio.

An engine’s maximum power then depends basically on how many revolutions per minute (RPM) it can or does produce. The greater the RPM, the more power strokes you have, and thus the more power the engine produces. Therefore, it makes sense that the most powerful engines generally also possess the highest RPM.

Because a piston with a short stroke doesn’t have to travel as far each cycle, it can cover a greater distance in the same amount of time versus an engine with a longer stroke and smaller bore. This means greater RPM and power output. Similarly, a larger bore means larger valves, which means it can pull in and push out more air and fuel mixture in each cycle, which again means more power.

This works in the opposite direction as well. Let’s say the goal is to gain efficiency rather than power. The best engine to use would be one with a small bore and a long stroke. This is a bit more complicated than the power equation as it involves the surface area. Basically, the more surface area a cylinder has during its combustion cycle, the less energy is lost to heat, resulting in a more efficient cycle.

The bore to stroke ratio is an oft cited statistic of engines and is determined by dividing the stroke length, in inches, into the bore diameter, also in inches, for a single cylinder. Thus, an engine with a 4-inch bore and 3.5-inch stroke would have a bore/stroke ratio of: 4/3.5 inches = 1.14.

The ratio of the cylinder volume at bottom dead center (BDC) to the cylinder volume at top dead center (TDC) is known as the compression ratio (Figure 2). It is influenced by the engine bore and stroke. A higher compression ratio leads to better combustion efficiency and increased power output.

Diesel engines use higher compression ratios than gasoline engines in that the lack of ignition using a spark plug means the compression ratio must increase the temperature of the air in the cylinder sufficiently to thermodynamically ignite the diesel-air mixture using ignition through compression. Compression ratios are often between 14:1 and 23:1 for direct injection diesel engines, and between 18:1 and 23:1 for indirect injection diesel engines, whereas gasoline engines are typically between 8:1 and 12:1. The compression ratio may be higher in engines running exclusively on liquified petroleum gas (LPG or propane) or compressed natural gas due to the fuel’s higher octane rating.

Engines are often classified as to their square characteristics, and generally defined as oversquare, square, or undersquare. A square engine has equal bore and stroke dimensions, giving a bore/stroke value of exactly 1:1 (Figure 1). The most common industrial engine is an oversquare type.

To determine an engine’s displacement, the area of the cylinder bore must first be determined. This is accomplished by squaring the bore diameter and then multiplying by 0.7854 (π/4) to derive the area. This value multiplied by the stroke length results in the displacement of a single cylinder. The total displacement of an engine is thereafter defined as:

Equation 1: Displacement (in in 3) = 0.7854 × Bore2 (in inches) × Stroke (in inches) × Number of cylinders

(SI units-cm3) = 0.7854 × Bore2 (in cm) × Stroke (in cm) × Number of cylinders

SI conversions: 1 liter = 1000 cubic centimeters (1000 cm3); 1 cubic inch = 16.387 cm3 = 0.01638 liter

Example 1: The bore diameter on an 8-cylinder engine is 4.312 inches; the stroke length is 3.65 inches. Find the total displacement (in cubic inches):

Solution: 0.7854 × (4.312 inches × 4.312 inches) × 3.65 inches × 8 cylinders = 426.4 cubic inches

Example 2: Using SI units: If the bore is 10 cm (3.93 inches), the stroke is 5 cm (1.97 inches) with four cylinders, the displacement calculation now becomes:

Solution: 0.7854 × (10 cm)2 × 5 cm × 4 cylinders = 1570 cm3 = 1.57 liters (95.8 ci) (Note: 1 liter = 1000 cm3)

Figure 3. Piston rings.

Three piston rings typically fit around the piston (Figure 3), bridging the small clearance between the piston and cylinder wall. The three piston rings perform different functions and are located at different spots on the piston.

The top and second rings are referred to as compression rings and are used to prevent the escape of gases between the piston and cylinder wall during the power stroke. The compression rings also act to transfer heat from the piston to the cylinder wall, where it is dissipated into the coolant flowing through the water jackets.

The lower ring on a piston is called an oil control ring. Oil is constantly sprayed onto the cylinder walls either from holes in the connecting rods or by jets installed in the crankcase. For minimal friction, a thin oil film is needed on surfaces of the cylinder wall and piston; thus, the function of the oil control ring is to remove excess oil and leave an ideal oil film for the compression rings and piston skirt to frictionlessly glide over.

Table 1 provides common bore and stroke dimensions along with horsepower output and approximate fuel consumption for many diesel and gasoline engines. These values are based on single cylinder performance. Therefore, to determine the total engine displacement and performance, the following values must be multiplied by the number of cylinders in the engine.

For example, a 53.4 CID single cylinder, multiplied by 8 cylinders will equal a total displacement of 53.4 CID × 8 = 427.2 CID with 18.19 HP/cylinder × 8 = 145.52 output HP at 1800 RPM.

### Mean Effective Pressure (MEP)

The term mean effective pressure (MEP) is a quantity relating to the operation of reciprocating engines and is a valuable measure of an engine’s capacity to do work that is independent of engine displacement. Generally, there are five distinct values of the mean effective pressure:

• Brake mean effective pressure (BMEP)
• Gross indicated mean pressure (IMEPG)
• Net indicated mean effective pressure (IMEPN)
• Pumping mean effective pressure (PMEP)
• Friction mean effective pressure (FMEP).

By using the brake, gross, and net mean effective pressure values, the pumping and friction values can be computed. When quoted as an indicated mean effective pressure or IMEP, it may be thought of as the average pressure acting on a piston during the different portions of the cycle.

The brake mean effective pressure (BMEP) is an effective measure used for comparing the performance of an engine of a given type to another of the same type and for evaluating the reasonableness of performance claims or specifications. The definition of BMEP is the average (mean) pressure, which if imposed on the pistons uniformly from the top to the bottom of each power stroke, would produce the measured (i.e., brake) power output in kW or HP.

Note that BMEP is purely a theoretical value derived from the engine’s torque and is used for estimating engine horsepower and has no association with the actual cylinder pressures. Expressed in a different way, the BMEP (in psi) multiplied by the piston area (in square inches) provides the theoretical value of the mean force applied to the piston during the power stroke.

Depending on the speed, diesel engines generally produce between 100 to 200 pounds of force per square inch. Multiplying that force by the stroke (in feet or inches divided by 12), gives the net product of work (in foot-pounds) produced by the piston moving from the top dead center to bottom dead center position (i.e., total travel distance of piston) with the BMEP exerted on the piston throughout that motion.

Typically, for a natural-aspirated, spark ignition, gasoline-fueled, two-valve-per-cylinder, pushrod engine, a BMEP over 200 psi (13.8 bar) is quite difficult to achieve and requires a serious development program and specialized components. On the other hand, a normal-aspirated compression-ignition diesel engine can easily produce over 220 psi (15.17 bar) of BMEP, and some turbocharged diesel engines can routinely exceed 300 psi (20.7 bar). Strictly as an estimate, although low for many diesel engines, the performance calculations in this column have been universally conducted using 150 psi as the BMEP value.

Another value of the mean effective pressure (MEP) is the indicated mean effective pressure (IMEP). As opposed to the BMEP, the IMEP is not a theoretical value and is considered as the actual in-cylinder pressure over the compression and expansion portion of the cycle (gross IMEP or IMEPG) or the cylinder pressure throughout the entire engine cycle (net IMEP or IMEPN).

The PMEP is determined from: PMEP = IMEPG – IMEPN, with the FMEP = IMEPN – BMEP. Although the BMEP and IMEP are different values, for the purposes of this column they are used interchangeably. When not readily available, the following equation can be used to estimate the BMEP from the engine’s output torque and displacement (in cubic inches or CID):

Equation 3: BMEP (for 4-stroke engines)
= 150.8 × Torque (in ft.-lbs.)
____________________ (For 2-stroke, use 75.4) Engine displacement (in CID)

Example 3: Find the estimated BMEP for a 4-stroke, 327 CID gas engine that produces 325 ft-lb. of torque:

Solution: Using Equation 3: 150.8 × 325 ft-lb./327 CID = 49,010/327 = 149.87 psi

### Fundamental Engine Horsepower and Torque Equations

It is always important to remember that the horsepower output from an engine relies on various factors including the bore diameter and stroke length, compression ratio, age and wear, engine design, fuel type and energy value, fuel consumption (BSFC), number of cylinders, number of power strokes per minute (RPM), and the cylinder’s ignition pressure (BMEP).

Therefore, prudent caution must be observed when using the following formulas for estimating engine HP. Engine horsepower is usually determined from torque because torque is easier to measure. Torque is defined specifically as a rotating force that may or may not result in motion. It’s measured as the amount of force multiplied by the length of the lever through which it acts in foot-pounds (ft.-lbs.).

For example, if a 1-foot-long wrench is used to apply 10 pounds of force to a bolt head, the wrench generates 10 foot-pounds of torque. Torque is an important component of engine sizing as it determines the engine’s ability to overcome the load’s internal drag, inertia, and friction to rotate and is often used to determine the horsepower output of an engine.

Disregarding internal friction and horsepower lost to heat, the following formula, often called the PLAN formula, is used to estimate an indicated HP output with each variable (P-LA-N) shown in italic with the second equation using torque and rotative speed, in RPM:

Equation 4a:
Indicated HP = P × L × A × N × (Number of cylinders)
__________
33,000

Equation 4b: Indicated HP = Torque (in ft.-lbs.) × RPM
__________________
5252

where:
P = Indicated Mean Effective Pressure (IMEP) or brake
effective mean pressure (BMEP) in psi (Note: When IMEP or BMEP is indicated in bars, convert to psi by multiplying bars × 14.50.)

L = Length of Stroke, in feet

A = Effective Area of Piston (bore) in square inches

N = Number of power strokes per minute (for two cycles use the RPM; for four cycles use RPM divided by 2)

Example 4a: Find the indicated HP (IHP) for a 327 CID, 8-cylinder, 4-stroke engine with an observed IMEP of 150 psi during engine operation at 3600 RPM; each cylinder has a 4-inch bore diameter and 3.25-inch stroke length:

Solution: Bore: 4″/2 = 22 × 3.14 = 12.56 in2
Stroke Length (ft.): 3.25″/12 = 0.2708′—N: 3600 RPM/2 = 1800
Indicated HP = 150 psi × 0.2708′ × 12.56 in2 × 1800 =
_________________________

33,000

918,337
_______= 27.828 HP per cylinder × 8 = 222.63 IHP
33,000

Example 4b: Find the estimated horsepower output for an engine that produces 320 ft.-lbs. of torque at 2400 RPM:

Solution: Using Equation 4b:
IHP = 320 ft-lbs. × 2400 RPM =   768,000
________________    _________= 146.23 IHP
5252                            5252

To estimate an older, generic gas engine’s single cylinder, 4-stroke HP using the bore diameter, stroke length, and RPM, Equation 4c can be used (only good for low compression engines and typically accurate to +/–20%):

Equation 4c: Estimated Engine HP = [(B)2 × S × RPM] ÷ 18,000

where: B = Bore diameter in inches

S = Stroke length in inches

### Direct Comparison of Engine Operating Unit Cost Versus Electrical Unit Cost

Figure 4. Typical range of thermal efficiency based on engine speed.

The subject of comparing the operating cost of an engine versus electrical driven component often arises. This is a calculation that requires working in SI units for simplicity to use kilowatts of power, the typical electrical unit of billing.

For this comparison, megajoules (MJ) and kilowatt-hours (kWh) are both units of energy where: 1 MJ = 0.278 kWh or 3.597 MJ = 1 kWh. Therefore, from Table 2, 1 liter (0.264 U.S. gallons) of diesel possesses 36 MJ, gasoline has 32 MJ, and LPG has 26 MJ of specific energy. Typical performance data can be obtained from Table 1. Thus, the equation to directly compare the energy unit cost of an engine to electrical kWh cost is:

Equation 5: Cost per kWh (at shaft)
= Cost of fuel per liter ×          1 × 3.597 (1/0.278)
_______________           ______________

Specific energy in MJ/liter      Engine efficiency (%)

Example 5: What is the cost per kWh for a diesel engine with 35% efficiency if the fuel cost is \$3.00 per gallon?

Solution: 1 gallon of diesel at \$3.00 = \$3.00 × 0.264 = \$0.792/liter
Cost per kWh = \$0.792 ×   1      ×    1
______    ___      ___
36                       0.35     0.278

= 0.022 × 2.857 × 3.597 = \$0.226/kW-hr.

The total operating cost per hour is obtained by multiplying the unit cost by the engine or motor kilowatts, which is the brake (BHP) or input horsepower (IHP) × 0.746.

### Fuel Consumption (BSFC) and Thermal Efficiency

The combination of the right bore and stroke can improve the engine’s thermal efficiency, which is the ratio of the useful work output to the energy input. The range of thermal efficiency for various speeds is illustrated in Figure 4. A well-designed engine with optimal bore and stroke can maximize efficiency and reduce fuel consumption.

Another common method of determining an engine’s efficiency is through use of the brake specific fuel consumption or its BSFC/SFC rating:

Equation 6: Fuel consumption estimate (SFC)
= Mass of Fuel Consumption         =   FC
_________________                    ____
Engine Brake Power                              Pb

where: SFC = specific fuel consumption

[(kg/h)/kW, kg/(3600 × kW), kg/(3600 kJ)]

FC = fuel consumption [lbm/hr]

Pb = brake horsepower [HP]

A typical BSFC value for a four-stroke natural-aspirated, liquid-cooled engine at 100% power output is between 0.42 to 0.46. A carburetor-equipped engine typically has a BSFC ranging from 0.48 to 0.55, while a fuel-injected engine tends to be slightly more efficient with a BSFC of around 0.45 to 0.50.

An average BSFC of 0.45 lbm/HP/hour is generally accepted for estimating purposes. Claims of gasoline engines with BSFC values less than 0.42 at maximum power delivery should be suspect. At reduced power settings (in the region of 70% and below) BSFC values as low as 0.38 have been achieved in practice and will likely become more commonplace as engine refinements evolve in the pursuit of lower fuel consumption and greater energy conservation.

Another method used to estimate the horsepower output of an engine is by tracking the fuel consumption in HP per gallon per hour of fuel consumption. See Table 3 for approximate values of fuel consumption for various sizes of naturally aspirated diesel engines:

Equation 7: For strict estimation purposes:
BSFC = 0.45 lbm/HP/hour -or- 5.92 × Fuel flow (in gph)
____________
Engine HP

Dividing this value into the water horsepower (WHP) × 100 yields the overall efficiency of the pumping plant. Refer to Example 6b.

Example 6: What is the estimated: (a) output HP and (b) pumping plant efficiency of an engine that delivers power to a well pump producing 665 GPM at 235 feet TDH with a coincidental fuel consumption of 7.6 GPH?

Solution: (a) HP: Using Equation 7: Engine Output HP (OHP) = (5.92)(7.6 GPH) ÷ 0.45 BSFC = 99.98 HP; (b) Plant
Efficiency: WHP = 665 GPM × 235′ TDH/3960 =39.46. Eff. = WHP/OHP = 39.46/99.98 = 0.395=39.5%

Until then, keep them pumping!

Ed Butts, PE, CPI, is the chief engineer at 4B Engineering & Consulting, Salem, Oregon. He has more than 40 years of experience in the water well business, specializing in engineering and business management. He can be reached at epbpe@juno.com.

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