Heat is extracted from hot rock at a temperature of 250°C and used to produce electricity with the ideal Carnot efficiency. If the temperature of the cold reservoir is $75 ^ { \circ } \mathrm { C }$, what mass of rock would be needed to yield $1 \mathrm { GWh } _ { e }$ ? See Chapter 8 for the thermal properties of typical rock, and assume a generator efficiency of $90 \%$.

Question

Quizplus · Accepted Answer

The Answer is From equation (8.11), the heat extracted from rock is

Q = C m \Delta T

From Table

8.6

, for rock

C = 879 \mathrm {~J} / \left( \mathrm { kg } \cdot { } ^ { \circ } \mathrm { C } ight)

. We need a net energy of

1 \mathrm { GWh } _ { \mathrm { e } }

and at

90 \%

efficiency the total energy required is

\left( 1 \mathrm { GWh } _ { \mathrm { e } } ight) / ( 0.9 ) = 1.1 \mathrm { GWh }

. Converting this to joules gives

( 1.1 \mathrm { GWh } ) imes \left( 10 ^ { 6 } \mathrm { kWh } / \mathrm { GWh } ight) imes \left( 3.6 imes 10 ^ { 6 } \mathrm {~J} / \mathrm { kWh } ight) = 3.96 imes 10 ^ { 12 } \mathrm {~J}

For the temperatures given for the hot and cold reservoirs in

{ } ^ { \circ } \mathrm { C }

, we calculate the equivalent temperatures in

\mathrm { K }

as

250 ^ { \circ } \mathrm { C } + 273 + 523 \mathrm {~K} ext { and } 75 ^ { \circ } \mathrm { C } + 273 = 348 \mathrm {~K}

Thus the ideal Carnot efficiency is

\eta = 1 - ( 348 \mathrm {~K} / 523 \mathrm {~K} )

)

= 0.335

(or

33.5 \%

). This means that the total energy required is

\left( 3.96 imes 10 ^ { 12 } \mathrm {~J} ight) / ( 0.335 ) = 1.18 imes 10 ^ { 13 } \mathrm {~J} ext {. }

For

\Delta T = 250 ^ { \circ } \mathrm { C } - 75 ^ { \circ } \mathrm { C } = 175 ^ { \circ } \mathrm { C }

, the mass is

m = \frac { Q } { C \Delta T } = \frac { 1.18 imes 10 ^ { 13 } \mathrm {~J} } { \left( 879 \frac { \mathrm { J } } { \mathrm { kg } \cdot { } ^ { \circ } \mathrm { C } } ight) imes \left( 175 ^ { \circ } \mathrm { C } ight) } = 7.3 imes 10 ^ { 8 } \mathrm {~kg}

Heat Is Extracted from Hot Rock at a Temperature of 250°C