A natural gas liquefaction plant produces liquid natural gas at $1$atm and $-161\mathrm{.}5C$ when natural gas enters at $1$atm and $20C.$ Treating the natural gas as methane, determine the minimum work input and maximum $($or reversible$)$ COP. If the liquefaction is done by a Carnot refrigerator between temperature limits of $T_H=20C$ and $T_L$ with the same reversible COP, determine the temperature $T_L.$

Various properties of methane before and after liquefaction process are given as follows:

$h1=-12\mathrm{.}12k\frac{J}{k}g$

$s1=-0\mathrm{.}040077k\frac{J}{k}g*K$

$h2=-910\mathrm{.}92k\frac{J}{k}g$

$s2=-6\mathrm{.}6767k\frac{J}{k}g*K.$

Answer: $T_L=135\mathrm{.}4K$

This is question $12-127$ in the ${10}^{th}$ edition of our book.