Chapter 4, Problem 88CP

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 ale as follows:

i. Isobaric intake stroke (OA). A mixture of gasoline and air is drawn into the combustion chamber at atmospheric pressure $P0$ as the piston expands, increasing the volume of the cylinder from zero to $V_A$ .

ii. Adiabatic compression stroke (AB). The temperature of the mixture rises as the piston

compresses it adiabatically from a volume $V_A$ to $V_B$ .

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 $V_B\left(=V_c\right)$ .

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 (DA). 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 $\left(i\right)e=W/Q_1;$

$\left(ii\right)w=Q_1-Q_2;$ and $\left(iii\right)Q_1=n{C}_v\left(T_C-T_B\right),Q_2=n{C}_v\left(T_D-T_A\right),$

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%.