A 35% efficient power plant puts out 920 MW of electrical power. Cooling towers are used to take

step 1 We can write the volume of the air heated in a per day is v over t, T for time, V for volume, is

A 35% efficient power plant puts out 920 MW of electrical...

A 35$\%$ efficient power plant puts out 920 $\mathrm{MW}$ of electrical power. Cooling towers are used to take away the exhaust heat. $(a)$ If the air temperature $\left(15^{\circ} \mathrm{C}\right)$ is allowed to rise 7.0 $\mathrm{C}^{\circ}$ , estimate what volume of air $\left(\mathrm{km}^{3}\right)$ is heated per day. Will the local climate be heated significantly? (b) If the heated air were to form a layer 150 $\mathrm{m}$ thick, estimate how large an area it would cover for 24 $\mathrm{h}$ of operation. Assume the air has density 1.2 $\mathrm{kg} / \mathrm{m}^{3}$ and that its specific heat is about 1.0 $\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{C}^{\circ}$ at constant pressure.

A 35$\%$ efficient power plant puts out 920 $\mathrm{MW}$ of electrical power. Cooling towers are used to take away the exhaust heat. $(a)$ If the air temperature $\left(15^{\circ} \mathrm{C}\right)$ is allowed to rise 7.0 $\mathrm{C}^{\circ}$ , estimate what volume of air $\left(\mathrm{km}^{3}\right)$ is heated per day. Will the local climate be heated significantly? (b) If the heated air were to form a layer 150 $\mathrm{m}$ thick, estimate how large an area it would cover for 24 $\mathrm{h}$ of operation. Assume the air has density 1.2 $\mathrm{kg} / \mathrm{m}^{3}$ and that its specific heat is about
1.0 $\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{C}^{\circ}$ at constant pressure.

Key Concepts

Density and Volume Relationship

The density of a substance links its mass to volume, allowing for energy calculations to be connected to changes in volume. When calculating the amount of air heated, the known density of air is used to convert between the volume of air and its corresponding mass, which is necessary to apply the specific heat capacity in energy calculations.

Energy Balance and Heat Transfer

This concept involves accounting for all energy entering and leaving a system to understand temperature changes and system behavior. For a given amount of waste heat and a known specific heat capacity of a medium (like air), one can determine the resulting temperature change. This is the basis for estimating how much mass of air needs to be heated to cause a certain temperature increase.

Waste Heat Rejection

In any energy conversion process, especially in thermal power plants, a portion of energy that is not converted into useful work is expelled as waste heat. Estimating the waste heat is essential for designing cooling systems and understanding environmental impacts, such as local air temperature changes due to heated exhaust gases.

Power Plant Efficiency

This concept involves the ratio of useful output energy (in this case, electrical power) to the total input energy provided to the system. In thermal systems, not all the input energy is converted into work; a significant fraction is rejected as waste heat. Understanding efficiency helps determine how much energy must be managed or dissipated, such as via cooling towers in power plants.

Specific Heat Capacity

This is the energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). It is a key property in thermal physics that allows the calculation of the amount of energy needed to produce a given temperature change in a material, which is crucial for analyzing heat transfer processes such as heating air in cooling towers.

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