Engine efficiency - using heat (kelvin)

[MikeNZ]

Homework Statement

A power plant taps steam superheated by geothermal energy to 505 K (the temperature of the hot reservoir) and uses the steam to do the work in turning the turbine of an electric generator. The steam is then converted back into water in a condenser at 323 K (the temperature of the cold reservoir), after which the water is pumped back down into the ground where it is heated again. The output power of the generator is 84 MW. Calculate:

1) The maximum efficiency at which this plant can operate
2) The minimum amount of rejected heat (in MJ) that must be removed from the condenser every 24 hours
3) Express your answer to (2) in TJ and in PJ.

(Quick question - what would TJ and PJ mean? Terrajoules and Petajoules or something?

Homework Equations

The Attempt at a Solution

For (1) - Can I calculate the ratio of heat going in, to heat going out.. and then use that ratio to the 84 MW to get the maximum efficiency?

I'm thinking for number (2) - Since 84 MW is power - meaning there's a time.. you can calculate the energy (without the time) and then apply that over the 24 hours (still trying to find the right equation) - I'm really stuck on number 2.

Number (3) - Have no idea as we have not been taught that - If it is Terrajoules and Petajoules, then it must be a simple case of moving the decimal point.

 

[Mapes]

Hi MikeNZ, welcome to PF. (Please don't post the same question in multiple places.)

The maximum efficiency will be that of a Carnot cycle, and this efficiency depends only on the temperature of the hot and cold reservoirs.

Your approach to #2 looks fine. TJ and PJ are indeed terajoules and petajoules.

 

[Mene]

However, I would like to know the significance of these units and their context in this problem.

I would like to provide a response to the content by breaking down the problem and providing a solution based on scientific principles and equations.

Firstly, the problem describes a power plant that uses geothermal energy to produce electricity. The hot reservoir, with a temperature of 505 K, provides the energy for the steam to turn the turbine and generate electricity. The steam is then cooled in the condenser, with a temperature of 323 K, before being pumped back into the ground to be reheated.

1) To calculate the maximum efficiency at which this plant can operate, we can use the Carnot efficiency formula, which is given by ƞ = 1 - (Tc/Th), where ƞ is the efficiency, Tc is the temperature of the cold reservoir, and Th is the temperature of the hot reservoir. In this case, the efficiency would be 1 - (323 K/505 K) = 0.36 or 36%.

2) The minimum amount of rejected heat that must be removed from the condenser every 24 hours can be calculated by using the equation Q = Pt, where Q is the heat, P is the power output (84 MW), and t is the time (24 hours). This gives us a value of 2016 MJ. This means that 2016 MJ of heat must be removed from the condenser every 24 hours to maintain the efficiency of the power plant.

3) TJ and PJ are indeed Terrajoules and Petajoules, which are units of energy. To convert MJ to TJ, we can divide by 1000, and to convert MJ to PJ, we can divide by 1,000,000. Therefore, the minimum amount of rejected heat in TJ would be 2.016 TJ, and in PJ would be 0.002016 PJ.

In conclusion, by using scientific principles and equations, we were able to calculate the maximum efficiency, minimum rejected heat, and express the answer in different units. This problem highlights the importance of understanding and utilizing thermodynamic principles in energy production and efficiency.