Evaluating Work

Measured data for pressure versus volume during the expansion of gases within the cylinder of an internal combustion engine are given in the table below. Using data from the table, complete the following:

(a) Determine a value of n such that the data are fit by an equation of the form, pVn = constant.

(b) Evaluate analytically the work done by the gases, in kJ, using Eq. 2.17along with the result of part (a).

(c) Using graphical or numerical integration of the data, evaluate the work done by the gases, in kJ.

(d) Compare the different methods for estimating the work used in parts (b) and (c). Why are they estimates?

Data Point |
p (bar) |
V (cm3) |

1 |
15 |
300 |

2 |
12 |
361 |

3 |
9 |
459 |

4 |
6 |
644 |

5 |
4 |
903 |

6 |
2 |
1608 |

Eq. 2.17

Step 1 of 4</p>

Part a

From the data given, we are required to calculate the value of from the equation constant.

Given:

constant

Taking logarithm on both sides,

, is the constant here

…..(1)

Comparing this equation with a straight line equation, will be the slope. If we plot the given data, we can calculate the slope and then the value of .

Data Point |
p (bar) |
V (cm3) |

1 |
15 |
300 |

2 |
12 |
361 |

3 |
9 |
459 |

4 |
6 |
644 |

5 |
4 |
903 |

6 |
2 |
1608 |

Convert the above values of and to logarithmic values and we get the following table.

log (V) log (p)

2.47712 1.17609

2.55751 1.07918

2.66181 0.95424

2.80889 0.77815

2.95569 0.60206

3.20629 0.30103

Note that is taken as the x-axis and log (p) as the y-axis and we get the following plot.

The vs plot is linear fitted in Origin Pro software. The fitting gives us the slope of -1.19 which is simply the value of .

Therefore, the value of is 1.19.

Step 2 of 4</p>

Part b

Equation 2.17 can be written as,

…..(1)

From the table,

bar N/m2

bar N/m2

cm3 m3

m3

Substitute these values in equation (1),

J

J

kJ

This is the required work done.