Question

For the Otto cycle of air shown, P1= 1 bar, T1 = 290 K, V1 = 400
cm3. If T3 = 2200 K and compression ratio is 8, find (a) the heat addition, in kJ, (b) the net work, in kJ, (c) the thermal efficiency.
The mass of the air is 0.48 g.

Answer 1

(a)

$Q_1=\nu C_V(T_3-T_2)$

$\gamma=1.4$

$n=\frac{V_1}{V_2}=8$

$C_V=\frac{R}{\gamma-1}$

$\nu=\frac mM$

$p_2=p_1(\frac{V_1}{V_2})^\gamma =p_1n^\gamma$

$T_2=T_1\cdot \frac{p_2V_2}{p_1V_1}=T_1 \cdot \frac{n^\gamma}{n}=T_1n^{\gamma-1}=290\cdot8^{0.4}=666 K$

$Q_1=\frac{0.48}{29}\cdot \frac{8.31}{0.4}\cdot(2200-666)=0.527 kJ$

(b)

$Q_2=\nu C_V(T_4-T_1)$

$p_3=p_2\frac{T_3}{T_2}=p_1n^\gamma\frac{T_3}{T_2}$

$p_4=p_3(\frac{V_2}{V_1})^\gamma=p_1n^\gamma\frac{T_3}{T_2}n^{-\gamma}=p_1\frac{T_3}{T_2}$

$T_4=T_3\frac{p_4V_4}{p_3V_3}=T_3\frac{p_1\frac{T_3}{T_2}}{p_1n^\gamma\frac{T_3}{T_2}}n=T_3n^{1-\gamma}=2200\cdot 8^{-0.4}=958 K$

$Q_2=\frac{0.48}{29}\cdot \frac{8.31}{0.4}\cdot(958-290)=0.230 kJ$

(c)

$\eta=1-\frac{1}{n^{\gamma-1}}=1-\frac{1}{8^{0.4}}=56$ (%)