A power plant extracts energy from steam at \(250^{\circ} \mathrm{C}\) and delivers 800 MW of electric power. It discharges waste heat to a river at \(30^{\circ} \mathrm{C} .\) The plant's overall efficiency is \(28 \% .\) (a) How does this efficiency compare with the maximum possible at these temperatures? (b) Find the rate of waste-heat discharge to the river. (c) How many houses, each requiring \(18 \mathrm{kW}\) of heating power, could be heated with the waste heat from this plant?

Waste heat is a byproduct of many industrial processes, particularly in power generation. In a thermal power plant, energy conversion is never 100% efficient, and some of the energy input is inevitably lost as waste heat. This waste heat is often released into the environment, commonly into water bodies such as rivers or lakes, which can act as cooling agents.

For example, imagine a power plant that operates at an efficiency of 28%, as in our exercise. This means that for every unit of energy it produces, a significant amount of energy is not converted into electricity but rather discharged as heat. This waste heat could be used for other purposes, such as heating buildings or powering additional processes, rather than being discharged. The energy not utilized in the process contributes to the overall thermal pollution of the natural water source, and its management is a crucial aspect of environmental conservation.

The concept of energy conversion efficiency is key in assessing the performance of a thermal power plant. It refers to the ratio of useful energy output to the total energy input. In the context of a power plant, the useful output is electrical energy, while the input is usually from burning fossil fuels or nuclear reaction.

The maximum theoretical efficiency a heat engine can achieve is defined by the Carnot efficiency, which is based on the temperatures of the heat source and the heat sink. The formula provided in the exercise, \(η_c = 1 - (Tc/Th)\), where \({T_c}\) is the cold reservoir temperature and \({T_h}\) is the hot reservoir temperature, both in Kelvin, is used to calculate this. For real-world engines, the actual efficiency is always lower than Carnot efficiency due to various real-world factors such as friction, material limitations, and other inefficiencies.

Increasing energy conversion efficiency is one of the major challenges in energy production as it has a direct impact on economic cost and environmental footprint. The lower the efficiency, the more fuel is required to produce the same amount of power, leading to increased costs and more waste heat.

A thermal power plant generates electricity by converting heat energy into electrical energy. The typical process involves a heat source to produce steam, which then drives a turbine connected to a generator. The temperatures mentioned in our exercise, with steam at \(250^{\text{\textdegree}} C\) and discharge temperatures at \(30^{\text{\textdegree}} C\), are indicative of the hot source and cooling medium, respectively.

A key concern in thermal power plants is the environmental effects of waste heat discharge. Though some of the waste heat is an unavoidable byproduct of the energy conversion process, finding ways to reuse this heat can significantly benefit both the power plant's efficiency and the environment. For instance, cogeneration or combined heat and power (CHP) systems can utilize this waste heat for heating purposes, as illustrated in the example of heating houses with the plant's excess heat.

The exercise's power plant operates at 28% efficiency and can significantly benefit from technologies like CHP. By utilizing the waste heat, such a plant could reduce fossil fuel consumption and greenhouse gas emissions, making it a more sustainable and economic option by providing not only electricity but also heating solutions to nearby communities.