The gasoline internal combustion engine operates in a cycle consisting of six
parts. Four of these parts involve, among other things, friction, heat
exchange through finite temperature differences, and accelerations of the
piston; it is irreversible. Nevertheless, it is represented by the ideal
reversible Otto cycle, which is illustrated below. The working substance of
the cycle is assumed to be air. The six steps of the Otto cycle are as
follows:
i. Isobaric intake stroke (OA). A mixture of gasoline and air is drawn into
the combustion chamber at atmospheric pressure \(p_{0}\) as the piston expands,
increasing the volume of the cylinder from zero to \(V_{A}\)
ii. Adiabatic compression stroke \((A B) .\) The temperature of the mixture
rises as the piston compresses it adiabatically from a volume \(V_{\mathrm{A}}\)
to \(V_{\mathrm{B}}\)
iii. Ignition at constant volume (BC). The mixture is ignited by a spark. The
combustion happens so fast that there is essentially no motion of the piston.
During this process, the added heat \(Q_{1}\) causes the pressure to increase
from \(p_{B}\) to \(p_{C}\) at the constant volume
\(V_{\mathrm{B}}\left(=V_{\mathrm{C}}\right)\)
iv. Adiabatic expansion (CD). The heated mixture of gasoline and air expands
against the piston, increasing the volume from \(V_{C}\) to \(V_{D}\) This is
called the power stroke, as it is the part of the cycle that delivers most of
the power to the crankshaft.
v. Constant-volume exhaust \((D A)\). When the exhaust valve opens, some of the
combustion products escape. There is almost no movement of the piston during
this part of the cycle, so the volume remains constant at
\(V_{A}\left(=V_{D}\right)\) Most of the available energy is lost here, as
represented by the heat exhaust \(Q_{2}\)
vi. Isobaric compression (AO). The exhaust valve remains open, and the
compression from \(V_{A}\) to zero drives out the remaining combustion products.
(a) Using (i) \(e=W / Q_{1} ;\) (ii) \(W=Q_{1}-Q_{2} ;\) and (iii) \(Q_{1}=n
C_{\nu}\left(T_{C}-T_{B}\right), Q_{2}=n C_{\nu}\left(T_{D}-T_{A}\right),\)
show that \(e=1-\frac{T_{D}-T_{A}}{T_{C}-T_{B}}\)
(b) Use the fact that steps (ii) and (iv) are adiabatic to show that
\(e=1-\frac{1}{r^{\gamma-1}}\) where \(r=V_{A} / V_{B}\) The quantity \(r\) is
called the compression ratio of the engine.
(c) In practice, \(r\) is kept less than around 7 . For larger values, the
gasoline-air mixture is compressed to temperatures so high that it explodes
before the finely timed spark is delivered. This preignition causes engine
knock and loss of power. Show that for \(r=6\) and \(\gamma=1.4\) (the value for
air), \(e=0.51,\) or an efficiency of \(51 \%\) Because of the many irreversible
processes, an actual internal combustion engine has an efficiency much less
than this ideal value. A typical efficiency for a tuned engine is about \(25
\%\) to \(30 \%\)
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