You have been tasked by seismic energy to assess the lifetime thermodynamic efficiency of an engineered geothermal system they installed one year ago in southern California. A geological engineering firm has reported that the time evolution of the temperature of the geothermal formation which you are extracting energy from will follow the general relationship:
T(t)=Ae-Bt + C
where T(t = 0 years)= 800K, T(t = 1 years)=790K, and T(t->8)->300K.
a. Solve for each of the constants A, B, and C and plot temperature versus time for 50 years.
b. Derive an expression for the maximum theoretical efficiency of a heat engine utilizing this heat source versus time. Plot the maximum theoretical efficiency versus time for 10 years. Assume that there is sufficient ocean water available to maintain a cold temperature of 295K for heat rejection.
c. Now suppose that the company has installed a steam Rankine heat cycle to convert this heat source into usable work. Plot the efficiency on the same graph as (b). The pressure ratio is 2 and the high pressure is 1 bar (i.e. the cycle runs under vacuum), the isentropic pump efficiency is 90%, and the isentropic turbine efficiency is 75% (Hints: For a basis assume a mass flow rate of 1kg/s of working fluid.