Question

Efficiency of a Carnot engine is $26.81\%$ . To increase its efficiency by $20\%$ , its sink temperature will have to be ____. The hot reservoir is maintained at $100^{\circ }C$ .

• A. increased by $20K$
• B. decreased by $30K$
• C. increased by $30K$
• D. decreased by $20K$

Answer 1

Answer: (D)

Efficiency of the heat engine is,
$\eta =\left(1-\frac{T_{cold}}{T_{hot}}\right)$
First case :
Let $T_{2i}$ be the temperature of cold reservoir in the first case.
$\frac{26.812}{100}=\left(1-\frac{T_{2i}}{373}\right)$
$\Rightarrow T_{2i}=\frac{73.19\times 373}{100}=273K$
Second case:
Final efficiency $=\frac{20}{100}\times 26.81+26.81$
$=32.172$
Let $T_{2f}$ be the temperature of sink in the second case,
$\frac{32.172}{100}=1-\frac{T_{2f}}{T_1}$
$\frac{T_{2f}}{373}=1-\frac{32.172}{100}$
$T_{2f}=373\left(1-\frac{32.172}{100}\right)$
$T_{2f}=253K$
$T_{2i}-T_{2f}=20K$