Chapter 4, Problem 89CP

An ideal diesel cycle is shown below. This cycle consists of five strokes. In this case, only air is drawn into the chamber during the intake stroke OA. The air is then compressed adiabatically from state A to state B, raising its temperature high enough so that when fuel is added during

the power stroke BC, it ignites. After ignition ends at C, there is a further adiabatic power stroke CD. Finally, there is an exhaust at constant volume as the pressure drops from $p_D$ to $p_A$ , followed by a further exhaust when the piston compresses the chamber volume to zero.

(a) Use $W=Q_1-Q_{2,}$ $Q_1=n{C}_p\left(T_C-T_B\right)$ , and $Q_2=n{C}_v\left(T_D-T_A\right)$ to show that

$e=\frac{W}{Q_1}1-\frac{T_D-T_A}{\gamma \left(T_C-T_B\right)}.$

(b) Use the fact that $A\to B$ and $C\to D$ are adiabatic to show that

$e=1-\frac{1}{\gamma }\frac{{\left(\frac{V_C}{V_D}\right)}^{\gamma }-{\left(\frac{V_B}{V_A}\right)}^{\gamma }}{\left(\frac{V_C}{V_D}\right)-\left(\frac{V_B}{V_A}\right)}.$

(c) Since there is no preignition (remember, the chamber does not contain any fuel during the compression), the compression ratio can be larger than that for a gasoline engine. Typically, $V_A/V_B=15$ and $V_D/V_C=5$ . For these values and $\gamma =1.4$ , show that $\delta =0.56$ , or an efficiency

of 56%. Diesel engines actually operate at an efficiency of about 30% to 35% compared with 25% to 30% for gasoline engines.