A home owner in an area with substantial elevation changes and a large annual rainfall (1.54 m) plans to supplement electricity needs of 1.1 kW average over the year with a hydroelectric installation. An open tank of 15 m in diameter collects precipitation that falls on its surface. Periodically the tank is drained through a penstock to a hydroelectric generator at an elevation of 100 m below the tank. Discuss the merits of this approach.

Question

Quizplus · Accepted Answer

The Answer is The total mass of water collected during the year is

( 3.14 ) imes ( 7.5 \mathrm {~m} ) ^ { 2 } imes ( 1.54 \mathrm {~m} / \mathrm { y } ) imes \left( 1000 \mathrm {~kg} / \mathrm { m } ^ { 3 } ight) = 2.72 imes 10 ^ { 5 } \mathrm {~kg} / \mathrm { y }

The maximum energy available (at

100 \%

efficiency) is

E = m g h = \left( 2.72 imes 10 ^ { 5 } \mathrm {~kg} ight) imes \left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } ight) imes ( 100 \mathrm {~m} ) = 2.67 imes 10 ^ { 8 } \mathrm {~J} / \mathrm { y }

The total energy use during the year is

\left( 1.1 imes 10 ^ { 3 } \mathrm {~W} ight) imes \left( 3.15 imes 10 ^ { 7 } \mathrm {~s} / \mathrm { y } ight) = 3.47 imes 10 ^ { 10 } \mathrm {~J} / \mathrm { y }

So the percentage of total energy use that can be supplied by the hydroelectric installation is

100 imes \left( 2.67 imes 10 ^ { 8 } \mathrm {~J} / \mathrm { y } ight) / \left( 3.47 imes 10 ^ { 10 } \mathrm {~J} / \mathrm { y } ight) = 0.77 \%

Not significant enough to be practical.

A Home Owner in an Area with Substantial Elevation Changes