Question

At steady state, a power cycle having a thermal efficiency of \(38 \%\) generates \(100 \mathrm{MW}\) of electricity while discharging energy by heat transfer to cooling water at an average temperature of \(70^{\circ} \mathrm{F}\). The average temperature of the steam passing through the boiler is \(900^{\circ} \mathrm{F}\). Determine (a) the rate at which energy is discharged to the cooling water, in Btu/h. (b) the minimum theoretical rate at which energy could be discharged to the cooling water, in Btu/h. Compare with the actual rate and discuss.

Short answer

Rate of energy discharged: 556,191,589.47 Btu/h. Minimum theoretical rate: 217,192,524.14 Btu/h.

Step-by-step solution

Determine electrical power output in Btu/h

Convert the electrical power output from MW to Btu/h. The conversion factor is 1 MW = 3,412,142 Btu/h. Thus, the electrical power generated is \( 100 \text{ MW} \times 3,412,142 \text{ Btu/h per MW} = 341,214,200 \text{ Btu/h} \).

Calculate the total energy input into the power cycle

Using the thermal efficiency formula \( \text{Efficiency} = \frac{\text{Electrical Power Output}}{\text{Total Energy Input}} \), the total energy input can be found: \( \frac{341,214,200 \text{ Btu/h}}{0.38} = 897,405,789.47 \text{ Btu/h} \).

Determine the rate of energy discharged to cooling water

The difference between the total energy input and the electrical power output is the energy discharged to the cooling water: \( 897,405,789.47 \text{ Btu/h} - 341,214,200 \text{ Btu/h} = 556,191,589.47 \text{ Btu/h} \). This is the rate at which energy is discharged to the cooling water.

Calculate the minimum theoretical energy discharge using Carnot Efficiency

For the Carnot efficiency, use the absolute temperatures: \( T_H = 900^{\text{°}F} + 459.67 = 1359.67^{\text{°}R} \)\( T_C = 70^{\text{°}F} + 459.67 = 529.67^{\text{°}R} \)\( \text{Carnot Efficiency} = 1 - \frac{T_C}{T_H} = 1 - \frac{529.67}{1359.67} = 0.611 \).

Calculate the minimum theoretical heat discharge rate

Using Carnot efficiency, find the minimum theoretical energy input, then the discharge: \( \frac{341,214,200 \text{ Btu/h}}{0.611} = 558,406,724.14 \text{ Btu/h} \). The heat discharged is \( 558,406,724.14 \text{ Btu/h} - 341,214,200 \text{ Btu/h} = 217,192,524.14 \text{ Btu/h} \).

Key concepts

steady state operation

In a power cycle operating at steady state, the conditions do not change over time. This means the amount of energy entering the system equals the amount of energy leaving over any given time period. Understanding steady state operation helps simplify calculations because we can assume consistent values for energy inputs and outputs.
In our problem, the power cycle operates at steady state, generating consistent electrical output while discharging a steady amount of energy to the cooling water.

• Energy in = Energy out
• Power cycles are common examples, like in power plants
• Consistency helps with predicting performance

By maintaining steady state operation, we can calculate the energy conversion and efficiency more accurately.

thermal efficiency calculation

Thermal efficiency is a measure of how well a power cycle converts heat into work (or useful energy). It is given by the formula:
\[ \text{Efficiency} = \frac{\text{Electrical Power Output}}{\text{Total Energy Input}} \times 100 \text{ (to express as percentage)} \]
In this exercise, the thermal efficiency is 38%, meaning 38% of the energy input is converted into electrical power, while the rest is likely wasted, usually as heat to the environment.
Using the provided 38% efficiency, we calculated:
\[ \frac{341,214,200 \text{ Btu/h}}{0.38} = 897,405,789.47 \text{ Btu/h} \text{ (Total Energy Input)} \]
This Total Energy Input helps track how efficiently our power cycle is performing and also helps predict energy losses.

energy conversion

Energy conversion is the process of changing one form of energy into another. In a power cycle, we're primarily concerned with converting thermal energy (from fuel, like steam) into electrical energy.
However, not all of the thermal energy gets converted. Some of it is discharged as waste heat, often to cooling water. The calculations in our problem explored this:

• Total energy input = 897,405,789.47 Btu/h
• Electrical power output = 341,214,200 Btu/h
• Energy discharged to cooling water = 556,191,589.47 Btu/h

This demonstrates the balance of energy in a power cycle: the total energy input minus useful output equals the energy that needs to be managed as waste.
To minimize waste, analyzing the minimum theoretical discharge using Carnot efficiency provides insight into what is achievable under ideal conditions.