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A water cooler of storage capacity $120$ litres can cool water at a constant rate of $P$ watts. In a closed circulation system (as shown schematically in the figure), tr e wat'r from the cooler is used to cool an external device that generates constantly $3 \mathrm{~kW}$ of heat (thermal load). The temperature of water fed into the device cannot exceed $30^{\circ} \mathrm{C}$ and the e.tire stored $120$ litres of water is initially cooled to $10^{\circ} \mathrm{C}$. The entire system is thermally insulat $\mathrm{d}$. The minimum value of $P$ (in watts) for which the device can be operated for $3$ hours is
(Specific heat of water is $4.2 \mathrm{~kJ}^{-1} \mathrm{~kg}^{-1}$ and the density of water is $10.$) $0 \mathrm{k}^2 \mathrm{~m}^{-3}$ )
$1600$
$2067$
$2533$
$3933$
Solution
Heat generated in $3 hrs =3(3600) 3 \times 10^3=324 \times 10^5 J$
Heat used by water heater $= ms \Delta T =120 \times 1 \times 4200 \times 2 O =100.8 \times 10^5 J$
Heat absorbed by coolant $= Pt =324 \times 10^5-100.8 \times 10^5=223.2 \times 10^5$
$P =\frac{223.2 \times 10^5}{3600}=2067 W$
