A geothermal generating facility operates at the ideal Carnot efficiency and
utilizes a pressurized hot water resource at 240°C.
(a) Calculate the Carnot efficiency as a function of the temperature of the cold reservoir from 40°C to 90°C.
(b) The waste heat from electricity generation may be utilized for space heating, and the facility may, therefore, supply a community with both electricity and heat. This approach is referred to as cogeneration and is discussed in more detail in Chapter 17. Using the relative proportions (averaged over the year) for residential electricity and heat requirements as presented in Chapter 2, is there an approach to regulating the cold
reservoir temperature that would make the best use of the geothermal resources?

Question

Quizplus · Accepted Answer

The Answer is (a) The Carnot efficiency is

\eta = 100 imes \left( T _ { h } - T _ { c } ight) / T _ { h }

So for

T _ { h } = 240 ^ { \circ } \mathrm { C } = 513 \mathrm {~K}

then the efficiency is

\begin{array}{lll}\hline T_{c}\left({ }^{\circ} \mathrm{C} ight) & T_{c}(\mathrm{~K}) & \eta(\%) \\hline 40 & 313 & 39.0 \45 & 318 & 38.0 \50 & 323 & 37.0 \55 & 328 & 36.1 \60 & 333 & 35.1 \65 & 338 & 34.1 \70 & 343 & 33.1 \75 & 348 & 32.2 \80 & 353 & 31.2 \85 & 358 & 30.2 \90 & 363 & 29.2\end{array}

(b) For residential users from Chapter 2, the total energy requirement for electricity and heat is

\begin{array} { l l } ext { heat } & 1.04 imes 10 ^ { 11 } \mathrm {~J} \ ext { electricity } & 4.32 imes 10 ^ { 10 } \mathrm {~J} \end{array}

for a total energy requirement of

1.47 imes 10 ^ { 11 } \mathrm {~J}

of which

100 imes \left( 4.32 imes 10 ^ { 10 } \mathrm {~J} ight) / \left( 1.47 imes 10 ^ { 11 } \mathrm {~J} ight) = 29.3 \%

is electricity. Thus, maintaining a higher cold reservoir temperature

\left( \sim 90 ^ { \circ } \mathrm { C } ight)

would best satisfy residential energy needs without the production of either excess heat or electricity.

A Geothermal Generating Facility Operates at the Ideal Carnot Efficiency