KEY 192A Learning Lab #4 Using Integrals in Real Life! November 6, 2017

Internal Combustion Engines Page 1

Primary Topic: Definite Integrals (MATH 160)

Supporting Topics: Work, Energy (PHYS 141); Internal Combustion Engines (MECH 337)

Technical Objectives:

·  Explain the physical interpretation of the definite integral as the area under a function, f(x).

·  Describe several examples from engineering practice wherein engineers use integrals to calculate areas under functions.

·  Perform integration of experimental data from a real engineering process using numerical integration.

1. The Definite Integral

In Calculus, you may have learned that the definite integral of a function f(x) over the interval [a, b] is written as follows:

(1)

The physical interpretation of the definite integral is that it represents the area under the curve f(x) over the interval a < x < b:

2. The Fundamental Theorem of Calculus

In some cases, it is possible to evaluate an integral analytically from a function’s anti-derivative using the Fundamental Theorem of Calculus, which states that:

“Let f(x) be continuous on interval [a,b] and let F be the antiderivative of f, then:

(2)

For example, if f(x) is the polynomial f(x) = x2 + 2, the definite integral of f(x) over the interval of 1< x < 3 can be evaluated as follows:

Can anybody think of an application where an engineer would be required to evaluate a definite integral in engineering practice?

In real life, engines use integrals all the time, although we rarely evaluate integrals analytically using the Fundamental Theorem of Calculus. Can you think of another way to evaluate the area under a curve of a function?

2. Internal Combustion Engines: A Real Application for Integrals!

In a spark ignited internal combustion engine, a premixed fuel-air mixture is compressed by decreasing the volume (V) of the gas in piston-cylinder by a factor of approximately 10. The high pressure mixture is then ignited by a spark plug and the pressure (P) increases rapidly. The high pressure gas then pushes down on the piston, which generates power as the gas expands back to its initial volume:

Compression Stroke Power Stroke

It is possible to instrument an internal combustion engine with a high speed pressure transducer to measure pressure vs. time for the entire cycle. Volume vs. time can be calculated from the geometry of the piston-cylinder as a function of the measured crank angle. The P and V data can then be plotted on against one another on a P-V diagram, also called an indicator diagram:

For each complete cycle, the net work [W, in Joules] produced by the gas within the piston-cylinder is equal to the net area of the P-V diagram:

(3)

where P is the pressure in N/m2 as measured by the pressure transducer and V is the volume in m3 as calculated from the geometry of the engine.

Note that area under the top curve (curve 3) on the PV diagram is work done BY the gas during the power stroke, whereas the area under the lower curve (curve 1) represents work done ON the gas during the compression stroke.

The indicated power [in Watts) of an internal combustion engine operating at a speed, N, is then calculated as follows:

(4)

The indicated power represents the maximum power that this particular engine under these operating conditions would generate if there were no other mechanical losses such as friction, etc.). Therefore, we use the indicated power to quantify the mechanical efficiency of an engine as follows:

(5)

Example 1. Evaluation of definite integrals in engineering practice: use of experimental P-V data to calculate indicated power of an internal combustion engine.

Known: The following experimental pressure and volume data were acquired from an internal combustion engine. The engine speed is 2500 RPM.

Find: The indicated work [J] per cycle and the indicated power [W] by finding the area enclosed in the PV diagram via “numerical integration”.

Solution: