Question

A promising method of power generation involves collecting and storing solar energy in large artificial lakes a few meters deep, called solar ponds. Solar energy is absorbed by all parts of the pond, and the water temperature rises everywhere. The top part of the pond, however, loses to the atmosphere much of the heat it absorbs, and as a result, its temperature drops. This cool water serves as insulation for the bottom part of the pond and helps trap the energy there. Usually, salt is planted at the bottom of the pond to prevent the rise of this hot water to the top. A power plant that uses an organic fluid, such as alcohol, as the working fluid can be operated between the top and the bottom portions of the pond. If the water temperature is \(35^{\circ} \mathrm{C}\) near the surface and \(80^{\circ} \mathrm{C}\) near the bottom of the pond, determine the maximum thermal efficiency that this power plant can have. Is it realistic to use 35 and \(80^{\circ} \mathrm{C}\) for temperatures in the calculations? Explain.

Short answer

Based on the given temperatures of the top and bottom portions of a solar pond power plant, and using the maximum (Carnot) efficiency formula, the calculated maximum thermal efficiency is approximately 12.74%. This value assumes that the given temperatures are realistic for this type of application, and in the analysis, it is determined that these temperatures are indeed plausible, considering the solar energy collection and insulation techniques in place. However, the actual efficiency would depend on the design and operational conditions of the power plant in practice.

Step-by-step solution

Convert temperatures from Celsius to Kelvin

To work with the thermodynamic formulas, we need to convert the temperatures given in Celsius to Kelvin. The conversion formula is: \(K = C + 273.15\) Using this formula, we convert the top temperature (\(35^{\circ} \mathrm{C}\)) and the bottom temperature (\(80^{\circ} \mathrm{C}\)) to Kelvin. \(T_{top} = 35 + 273.15 = 308.15\,\mathrm{K}\) \(T_{bottom} = 80 + 273.15 = 353.15\,\mathrm{K}\)

Calculate the maximum (Carnot) efficiency

The maximum (Carnot) efficiency of the power plant can be calculated using the following formula: \(\eta_{Carnot} = 1 - \frac{T_{low}}{T_{high}}\) Where: - \(\eta_{Carnot}\) is the maximum (Carnot) efficiency of the power plant - \(T_{low}\) is the lower temperature (top) in Kelvin - \(T_{high}\) is the higher temperature (bottom) in Kelvin Using the converted temperatures, we can now calculate the maximum (Carnot) efficiency: \(\eta_{Carnot} = 1 - \frac{308.15}{353.15} = 1 - 0.8726 = 0.1274\) Expressing the efficiency as a percentage: \(\eta_{Carnot} = 12.74\%\)

Determine the validity of chosen temperatures

Now we need to consider if our given temperature values of \(35^{\circ} \mathrm{C}\) and \(80^{\circ} \mathrm{C}\) are realistic for this type of application. Let's analyze each temperature separately. \(35^{\circ} \mathrm{C}\) is a reasonable temperature for the top part of the pond, as it can be achieved during daylight hours in hot climates. Infrared panels, heat retainers, or other techniques could be used to maintain this temperature during the night, too. \(80^{\circ} \mathrm{C}\) at the bottom of the pond might be realistic if adequate solar energy is collected and the insulation at the top of the pond is efficient. By having the right salt gradient concentration, hot water trapped at the bottom could potentially reach this temperature. Overall, it is realistic to use these temperature values in the calculations for a solar pond-based power generation system. However, in practice, the design and operational conditions of the system should be taken into account to determine the actual efficiency.