Homework #1 MECH.5350 – Fundamentals of Sustainable Energy (Due on February 11 on BB)

Homework #1 MECH.5350 – Fundamentals of Sustainable Energy (Due on February 11 on BB)

1. The Rankine vapor cycle (close-cycle, shown in the figure) is the basis for a widely used combustion cycle that uses coal, fuel oil, or other fuels to compress and heat water to vapor, and then expand the vapor through a turbine in order to convert heat to mechanical energy. The following stages occur between states in the cycle:

1. 1->2 Compression of the fluid using a pump

2. 2->3 Heating of the compressed fluid to the inlet temperature of turbine, including increasing temperature to the boiling point, and phase

change from liquid to vapor

3. 3->4 Expansion of the vapor in the turbine

4. 4->1 Condensation of the vapor in a condenser

Please draw schematic temperature-entropy diagram for the above cycle. Use equations to support your

work. (Hint: S   dQ ) T

2. In geothermal energy conversion, hot underground water is sometime available. As shown in the figure, heat source lies within a layer of water deep beneath the ground surface. The porosity, p’, is the fraction of the aquifer containing water, assuming the remaining space to be rock of density ρr. The minimum useful temperature is T1. Assume that the thickness of the aquifer h<<z2, therefore the water (and the rock in the aquifer) is all considered at temperature T2. Now derive the expression for the total heat content and time constant (with the removal of heat by a water with volume flow rate V at θabove T1).

Note: the total heat content should include the water and the rock in the aquifer, as well as the rock above them.

Hint: Thermal capacity of the aquifer is the average:

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