Question

A geothermal power plant uses geothermal water extracted at 160°C at a rate of 440 kg/s as the heat source, and produces 22 MW of net power. If the environment temperature is 25°C, determine (a) (0.5 point) the actual thermal efficiency, (b) (0.5 point) the maximum possible thermal efficiency, and (c) (0.5 point) the actual rate of heat rejection from this power plant

Answer 1

The actual thermal efficiency of a geothermal power plant is found by dividing the power output by the heat input, while the maximum possible thermal efficiency is based on the Carnot efficiency. The actual rate of heat rejection is the difference between the heat input and the power output. Drilling depths for higher temperatures are calculated based on the earth's thermal gradient.

Step-by-step explanation:

Geothermal Power Plant Efficiencies and Heat Rejection

For a geothermal power plant that utilizes geothermal water extracted at 160°C and operates at an environmental temperature of 25°C:

• (a) Actual thermal efficiency is calculated by dividing the net power output by the rate of heat input from the geothermal water. The heat input can be determined by the product of the mass flow rate of the geothermal water and its specific heat capacity (if we assume water has a specific heat capacity of 4.18 kJ/kg°C, which can vary slightly with temperature).
• (b) Maximum possible thermal efficiency is determined by the Carnot efficiency, which depends on the temperatures of the heat source and the environment. It is calculated as (1 - Tc/Th), where Tc is the cold temperature reservoir (environment) and Th is the hot temperature reservoir (geothermal water) in Kelvin.
• (c) Actual rate of heat rejection is the difference between the rate of heat input and the net power output of the plant. If the plant produces 22 MW and we've calculated the heat transfer in from (a), we subtract the 22 MW to find the rate at which heat is being rejected.

For a typical temperature gradient of 25°C per kilometer, the depth required for a geothermal plant to match 35% fossil fuel plant efficiency would be based on the thermal gradient and desired temperature difference to attain the effective working temperatures for the Carnot efficiency that would yield an actual efficiency matching the 35%. As for how deep we would have to drill, it would depend on the difference in temperature needed to achieve that efficiency, taking into account the temperature gradient.

Considering the rate of heat rejection, the environmental impact of such heat release should be carefully assessed. This heat, while a byproduct of the power generation process, could be potentially utilized in district heating or other applications, albeit it is not often economically viable.

Answer 2

The actual efficiency is:

`n_(actual) = 2,76 %`

The maximum efficiency possible is:

`n_(max) = 31,18 %`

The rate of heat rejection is:

`Q_(rej)  = 774621,3 kW`

Step-by-step explanation:

Considering the specific heat of water 4,1813 KJ/kgK the energy that goes in the system is:

`Q_(in) = 440 kg/s * 4,1813 KJ/kgK * 433 K = 796621,3 kW`

The heat energy that goes out in an ideal situation (Tout = 25°C):

`Q_(out) = 440 kg/s * 4,1813 KJ/kgK * 298 K = 548252,1 kW`

The maximum heat that can be obtained is:

`Q_(max)  = Q_(in) - Q_(out) = 248369,2 kW`

So the maximum efficiency possible is:

`n_(max) = (Q_(max) )/(Q_(in) ) = 31,18 %`

The actual efficiency is:

`n_(actual) = (W_(out) )/(Q_(in) ) = \frac{22000 kW} }{796621,3 kW } =2,76 %`

Where
`W_(out)` is the obtained work equal to 22 MW

From the balance of energy, the rate of heat rejection is:

`Q_(rej)  = Q_(in) - W_(out) = 774621,3 kW`