Question

Many gasoline, internal combustion, piston engines are based on the Otto cycle which can be broken down into four steps:

1) A fuel-air mixture in a cylinder fitted with a piston is compressed rapidly from volume V1 and pressure P1 to a volume V2 and pressure P2, at which point a spark ignites the fuel-air mixture.

2) The fuel burns very quickly, increasing the pressure from P2 to P3 in a fixed volume V2.

3) This increase in pressure forces the piston outward, increasing the volume back to the

original value V1 and reducing the pressure to P4.

4) At this point, a valve opens to reduce the pressure from P4 to P1 while keeping the volume

fixed at V1.

Note that a subsequent stroke is needed to expel the exhaust gases and fill the chamber with new fuel-air mixture to return the system to its starting point. However, we can ignore this part of the cycle as well as the details of Step 4 because the work involved with them is small. In an engine, the cycle is very fast so Steps 1 and 3 can be approximated as adiabatic. As well, the fuel-air mixture is often approximated as an ideal gas.

a) On a labelled P − V diagram, sketch the Otto cycle, indicating the values of P and V at the beginning and end of each of the four Steps. Label the four Steps, putting arrows to show the direction the cycle is traversed, and show graphically the total work done by the cycle.

b) Assuming the Otto cycle were performed reversibly, find expressions for q and w for Steps 1–3 in terms of the appropriate pressures, volumes, and CV,m, the latter of which can be approximated as constant throughout the cycle. Note that in principle the number of moles of gas could change in Step 2 depending upon the stoichiometry of the combustion reaction of the fuel. However, the fuel typically is in low concentration so ignore this effect and treat the number of moles as approximately constant in Steps 1–3.

c) Using the results from part b), show that the efficiency of the reversible Otto cycle is η=|w|/|q_h| =1-1/(r^γ−1 )

in which qh is the heat entering the cycle in Step 2, w is the total work of the cycle (from Steps 1 and 3), r = V1/V2 is the compression ratio, and γ = CP,m/CV,m is the ratio of heat capacities. Note: the result of Question 6 may prove helpful.

d) For a typical engine, γ ≈ 1.3 and r ≈ 10. Use the formula from part c) to calculate the efficiency. Do you expect this to be the efficiency of an actual gasoline engine? Please justify your response.

e) The result of part c) shows that the compression ratio is the main design characteristic that limits engine efficiency. Gasoline engines typically have r ≈ 10 but diesel engines (which use a cycle similar to the Otto cycle except they inject fuel directly into the cylinder at the end of the compression stroke) typically have r ≈ 15 − 23, making them in general more efficient. Why, in practice, cannot a typical gasoline engine be designed with a compression ratio comparable to a diesel engine?