Question

Common Data Question:

In two air standard cycles - one operating in the Otto and the other on the Brayton cycle - air is isentropically compressed from $300$ to $450,K$. Heat is added to raise the temperature to $600K$ in the Otto cycle and to $550K$ in the Brayton cycle.

In ${\eta }_o$ and ${\eta }_B$ are the efficiencies of the Otto and Brayton cycles, then

• A. ${\eta }_o=0.25,{\eta }_B=0.18$
• B. ${\eta }_o={\eta }_B=0.33$
• C. ${\eta }_o=0.5,{\eta }_N=0.45$
• D. It is not possible to calcualte the efficiencies unless the temperature after the expansion is given

Answer 1

The correct option is B ${\eta }_o={\eta }_B=0.33$





For Otto cycle
$T_1=300K;T_2=450K$
$T_3=600K$
For process $1-2$,
$\frac{T_2}{T-1}={\left(\frac{V_1}{V_2}\right)}^{\gamma -1}$ or
$\frac{V_1}{V-2}={\left(\frac{T_2}{T_1}\right)}^{\frac{1}{\gamma -L}}={\left(\frac{450}{300}\right)}^{\frac{1}{1.4-1}}$
$(1.5{)}^{2.5}=2.75$
i.e., Compression ratio,
$r=\frac{V_1}{V_2}=2.75$
$\therefore {\eta }_{Otto}=1-\frac{1}{r^{\gamma -1}}=1-\frac{1}{(2.75{)}^{1.4-1}}$
$=1-0.667=0.33=33$%
For Brayton cycle
$T_1=300K;T_2=450K$
$T_3=550K$
For process $1-2$,
Pressure ratio,
$r_p=\frac{p_2}{p_1}={\left(\frac{T_2}{T_1}\right)}^{\frac{\gamma }{\gamma -1}}$
${\eta }_{Brayton}=1-\frac{1}{(r^p{)}^{\frac{\gamma -1}{\gamma }}}=1-\frac{1}{{\left[{\left(\frac{T_2}{T_1}\right)}^{\frac{\gamma }{\gamma -1}}\right]}^{\frac{\gamma }{\gamma -1}}}$
$=1-\frac{1}{T_2/T_1}=1-\frac{T_1}{T-2}=1-\frac{300}{450}$
$=0.33=33$%
Calculatign work output in otto and Brayton cycle which is required for question $44$
also ${\eta }_{Otto}=\frac{\text{Work,output}:W_{Otto}}{\text{Heat,supplied}:Q_1}$
$0.33=\frac{W_{Otto}}{Q_1}$
or $W_{Otto}=0.33Q_1$
$=0.33\times c_v(T_3-T_2)$
$=35.54kJ/kg$
${\eta }_{Brayton}=\frac{\text{Work,output}:W_B}{\text{Heat,supplied}}$
$0.33=\frac{W_B}{Q_1}$
or $W_B=0.33Q_1$
$=0.33\times c_p(T_3-T_2)$
$0.33\times 1.005(550-450)$
$=33.165kJ/kg$
Hence work ouptut per unit mass of the Otto cycle is more than the work output per unit mass of the Brayton cycle.