Suppose that a deep shaft were drilled in Earth\'s crust near one of the poles,

Question

Suppose that a deep shaft were drilled in Earth's crust near one of the poles, where the surface temperature is -34

Suppose that a deep shaft were drilled in Earth's crust near one of the poles, where the surface temperature is -34 C, to a depth where the temperature is 811 C. What is the theoretical limit to the efficiency of an engine operating between these temperatures? If all the energy released as heat into the low-temperature reservoir were used to melt ice that was initially at -34 C, at what rate could liquid water at 0 C be produced by a 100 MW power plant (treat it as an engine)? The specific heat of ice is 2220 J/kg middot K; water's heat of fusion is 333 kJ/kg. (Note that the engine can operate only between 0 C and 811 C in this case. Energy exhausted at -34 C cannot warm anything above -34 C.) I correctly solved (a) as 78%, but I incorrectly solved (b) as 79.1 kg/s. Help me with b?

Explanation / Answer

The heat would have to be exhausted at 0 deg C, or 273 K
The hot reservoir would = 811 + 273 = 1084 K
The max eff of the powerplant would = 1 - [273 / 1048] = 0.739
The rate of work done would be 100 MJ persec so the heat input would be
100 / 0.739 = 135.32 MJ/sec
The heat output would be 135.32 - 100 = 35.32 MJ per sec
To melt one kg of ice, it would take
m c deltaT + mL
= 1 * 2220 * 34 + 1 * 333000 = 408480 Joules = 0.408480 MJ
it would produce liquid water at a rate of
35.32 MJ per sec * 1 kg / 0.408480 MJ
= 86.47 kg/sec

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