A geothermal scheme is proposed

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · The laws of thermodynamics · The laws of thermodynamics

Thermal Physics, Ch Heat engines, refrigerators and the Carnot cycle

Problem

A geothermal scheme is proposed. Pressurised brine supplies heat at a steady

T_h = 480\,\mathrm{K}

and waste heat is rejected to cooling water at

T_c = 300\,\mathrm{K}

; both bodies are large enough to count as reservoirs. The designer claims that the turbine loop, running in a cycle, absorbs

Q_h = 1.00\,\mathrm{kJ}

from the brine and delivers

W_{\rm out} = 450\,\mathrm{J}

of work each cycle. Asked about the Carnot bound, the designer replies that the bound was derived for an ideal gas, whereas the loop uses an organic fluid that boils and condenses, so its ratio

Q_h/Q_c

need not equal

T_h/T_c

. ((a)) Find the claimed efficiency and the efficiency of a reversible engine working between these two reservoirs. Is the claim possible? ((b)) Suppose the claimed engine did exist and its work output were used to drive a reversible engine backwards between the same two reservoirs. How much heat per cycle would the pair move from the cold reservoir to the hot reservoir, with no other effect? Which statement of the second law does that contradict? ((c)) Answer the designer's second point: give the value of

Q_h/Q_c

for any reversible engine between these reservoirs, and say why the working substance does not enter.

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