The Efficiencies of Real Engines — Question 5

Question 5

Constant-pressure heat addition changes the ideal cycle efficiency

An idealized Diesel cycle has reversible adiabatic compression 1 to 2 with ratio $r=V_1/V_2=16$, constant-pressure heating 2 to 3 with cutoff ratio $\rho=V_3/V_2=2$, adiabatic expansion 3 to 4 back to $V_1$, and constant-volume cooling 4 to 1. Use constant $\gamma=1.40$ and $C_p/C_V=\gamma$.

1. Derive $T_2/T_1$, $T_3/T_2$ and $T_4/T_1$, then the ratio of rejected to absorbed heat.

2. Calculate efficiency and compare with the ideal Otto formula at the same compression ratio. Explain why this is not a universal ranking of actual engine designs.

Question 5 - Solution

Constant-pressure heat addition changes the ideal cycle efficiency

(a) Method and calculation

$$T_2/T_1=r^{\gamma-1},\quad T_3/T_2=\rho,\quad T_4/T_1=\rho^\gamma.$$$$\frac{Q_c}{Q_h}=\frac{C_V(T_4-T_1)}{C_p(T_3-T_2)}=\frac{\rho^\gamma-1}{\gamma r^{\gamma-1}(\rho-1)}.$$

(b) Interpretation and further calculation

$$\eta_D=1-\frac{2^{1.4}-1}{1.4(16^{.4})(2-1)}\approx.613805.$$$$\eta_O=1-16^{-.4}\approx.670123.$$The comparison fixes compression ratio, not heat input, peak temperature, peak pressure or real losses. Changing those constraints changes the engineering comparison.

Physical check

The cutoff ratio must satisfy $1<\rho<r$ for the stated expansion leg.