Question

(heat engines) Let us propose that we use geothermal energy to drive a heat engine. Let us also assume that we find a geothermal well under the Pacific not too far off shore. We are going to extract hot water from the well (at 940 C) and expel the waste heat at the ocean's surface (at an average constant temperature of 14° C.) a) What is the maximum possible efficiency of a heat engine operating between these two temperatures? b) Roughly estimate the volume of water needed every second by a 2 GW power plant assuming ideal efficiency, your answer should be in m'/s.

Answer 1

T_hot = 94 degreeC = 94 + 273 = 367 k and T_cold = 14 degreeC = 287 k Now q lessthanorequalto 1 - T_cold/T_hot therefore Maximum efficiency isa: q = T_hot - T_cold/T_hot = 80/367 = 0.218 b) Heat Extracted from in given by: Q = m c_w delta T where m is mass of c_w is specific heat of water = 4186 J kg^-1 k6-1 delta T = T_hot - T_cold = {80 degree c OR 80 k However the efficiency of engine neta(< 1) i.e. neta theta amount of the heat theta extracted from water is converted into useful energy by the power plant. The power plane generates 2GW of power. If M is the mass of water extracted from the well every second, then neta M c_w delta T = 2 times 10^9 J s^-1 M = 2 times 10^9/neta c_w delta T Volume of water per recorded, V = M/p_w where p_w = clarity of water = 1000kg m^-3 V = 2 times 10^9