8. Geothermal energy. A very large mass M of porous hot rock is to be utilized t

Question

8. Geothermal energy. A very large mass M of porous hot rock is to be utilized to generate electricity bv injecting water and utilizing the resulting hot steam to drive a turbine. As a result of heat extraction, the temperature of the rock drops, according to ?Qh -MC dTh, where C is the specific heat of the rock, assumed to be temperature independent. If the plant operates at the Carnot limit, calculate the total amount W of electrical energy extractable from the rock, if the temperature of the rock was initially?=?, and if the plant is to be shut down when the temperature has dropped to Th - Tf. Assume that the lower reservoir temperature Tt stays constant. At the end of the calculation, give a numerical value, in kWh, for M = 1014 kg (about 30 km*), C-1 J g-1 K-1, Ti=600"C, Tf-110°C, Tl=20%C Watch the units and explain all steps! For comparison: The total electricity produced in the world in 1976 was between 1 and 2 times 1014 kWh

Explanation / Answer

at certain instance, temperature of the rock , let be, T

then carnot efficiency=(hot temperature-cold temperature)/hot temperature

=(T-Tl)/T

heat extracted =dQ=M*C*dT

work done=dW=heat extracted *efficiency

==>dW=M*C*(1-(Tl/T))*dT

integrating both sides,

W=M*C*T-M*C*Tl*ln(T)

using limits,

total electrical energy extracted=M*C*(Ti-Tf)-M*C*Tl*ln(Ti/Tf)

given numerical values are:

M=10^14 kg

C=1 J/(gram.K)=1 kJ/kg.K

Ti=600 degree celcius=600+273=873 K

Tf=110 degree celcius =110+273=383 K

Tl=20 degree celcius=273+20=293 K

then W=M*C*(Ti-Tf)-M*C*Tl*ln(Ti/Tf)

=10^14*1000*(873-383)-10^14*1000*293*ln(873/383)

=2.486*10^19 J

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