Num_Integration_example

Num_Integration_example - π d 2/4 = 50.265 in 2 i 1 2 3 4...

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Work done by Piston in a 2 stroke IC engine: Numerical Integration Prof. Dragan Djurdjanovic ME 218 Fall 2008

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Introduction The pressure acting on a piston in a 2-stroke IC engine is measured to determine the power characteristics of the engine. The pressure is measured at various piston positions from the Top Dead Center (TDC).
The Problem The piston has a diameter of 8 inches travels a stroke of 15 inches. Data is collected for discrete piston displacements away from TDC. Work is given as where, A = area of piston and p(x) is the pressure at distance ‘x’ inches from TDC Using the Trapezoidal numerical integration techniques, compute the work done as it travels a stroke of 15 inches 15 0 ) ( dx x p A

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The Data Piston diameter, d = 8 inches Area =

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Unformatted text preview: π d 2 /4 = 50.265 in 2 i 1 2 3 4 5 6 7 8 9 10 11 12 x i (in) 1 2 2.5 4 5 7 8 11 12 13 14 15 p(x i ) psi 110 130 140 170 185 210 215 185 135 100 90 70 40 Numerical Integration A look at the given data reveals uneven intervals of x Use Trapezoidal rule Integral = ∑ {0.5*(x i+1 – x i )*[ p(x i ) + p(x i+1 )]} where the summation is from i = 0 to 11 Final Solution Using the above formula, the numerical solution of the integral of p(x) with respect to x for x varying from 0 to 15 inches yields an answer of 2248.75 lb per in. Multiplying by the area of the piston gives the work over the stroke as 113033.4 lb – in Thus the work done over the stroke is 9419 lb – ft (around 12800 J) The End...
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Num_Integration_example - π d 2/4 = 50.265 in 2 i 1 2 3 4...

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