Question

A well-established way of power generation involves the utilization of geothermal energy-the energy of hot water that exists naturally underground-as the heat source. If a supply of hot water at \(140^{\circ} \mathrm{C}\) is discovered at a location where the environmental temperature is \(20^{\circ} \mathrm{C},\) determine the maximum thermal efficiency a geothermal power plant built at that location can have. Answer: 29.1 percent

Short answer

Answer: The maximum thermal efficiency of the geothermal power plant is approximately \(29.1\%\).

Step-by-step solution

Convert the temperatures to Kelvin

First, we need to convert the given temperatures to Kelvin, as the efficiency formula requires it. Adding \(273.15\) to the Celsius temperatures gives: - Hot water temperature: \(140^{\circ}\mathrm{C} + 273.15 = 413.15\,\mathrm{K}\) - Environmental temperature: \(20^{\circ}\mathrm{C} + 273.15 = 293.15\,\mathrm{K}\)

Calculate the maximum thermal efficiency

We can now use the formula for the maximum thermal efficiency of a heat engine, which is given by \(1 - \frac{T_L}{T_H}\), where \(T_H\) is the temperature of the high-temperature reservoir and \(T_L\) is the temperature of the low-temperature reservoir. Plugging in the temperatures we found: \(\mathrm{Thermal\;Efficiency} = 1 - \frac{293.15\,\mathrm{K}}{413.15\,\mathrm{K}} = 1 - 0.709 \approx 0.291\)

Convert the efficiency to a percentage

To express the thermal efficiency as a percentage, we multiply it by 100: \(0.291 \times 100 = 29.1\%\) Hence, the maximum thermal efficiency a geothermal power plant built at that location can have is approximately \(29.1\%\).

Key concepts

Maximum Thermal Efficiency

Understanding the maximum thermal efficiency of a power plant is crucial in determining how effectively it converts heat into work. This efficiency is inherently tied to the laws of thermodynamics, which dictate that no heat engine can be 100% efficient due to the inevitable loss of some energy to the surroundings.

In a geothermal power plant, the earth's heat is used to generate electricity, and the maximum thermal efficiency represents the highest possible ratio of electrical energy output to the thermal energy input from the geothermal source. The maximum efficiency is dependent on the temperatures of the heat source and the environment into which the plant exhausts heat, typically the ambient temperature.

To improve the student's grasp on this concept, it is important to emphasize that the maximum thermal efficiency is theoretical. It assumes a perfect, frictionless engine and doesn't take into account practical losses such as those due to the mechanics of the power plant, the properties of the working fluid, and other real-world factors. By understanding this, students can recognize why real-world efficiencies will always be lower than the theoretical maximum.

Kelvin Temperature Conversion

Temperature conversion to Kelvin is an essential step when calculating thermal efficiency because thermodynamic equations require an absolute temperature scale. Unlike Celsius or Fahrenheit, Kelvin is an absolute scale where 0 K indicates the absence of thermal energy, known as absolute zero.

To convert Celsius to Kelvin, one needs to add 273.15 to the Celsius temperature. This is because the Kelvin and Celsius scales are offset by 273.15 degrees—the difference in temperature between absolute zero and the freezing point of water. For instance, in the geothermal power plant exercise, the hot water temperature of 140°C converts to 413.15 K, and the environmental temperature of 20°C converts to 293.15 K. This conversion is vital for accurately calculating the thermal efficiency as it sets a common ground for comparison and calculations in a thermodynamics context.

When crafting explanations for students, it is advantageous to present real-world applications of Kelvin, like its use in physics and engineering, to demonstrate its significance beyond just a formula. Including examples or analogies may assist students in understanding the role of Kelvin in thermodynamics and other scientific fields.

Thermal Efficiency Formula

The thermal efficiency formula is a mathematical representation of how well a heat engine, such as a geothermal power plant, converts heat into work. The formula is expressed as 1 - T_L/T_H, where T_L is the absolute temperature of the low-temperature reservoir (often the environment), and T_H is the absolute temperature of the high-temperature reservoir (the heat source). This formula is derived from the Carnot cycle, which describes the maximum efficiency a heat engine can achieve with reversible processes.

In the context of the exercise, upon converting Celsius temperatures to Kelvin, we can apply the formula to find that the plant's maximum theoretical efficiency is 29.1%. It's important for students to note that this figure represents the ceiling of the plant's capability under ideal circumstances. The simplicity of the formula belies the complex considerations in real-life systems, which include irreversible processes and additional energy losses.

When teaching this concept, use a variety of examples to show how the efficiency formula can differ based on the temperatures involved. Encouraging students to experiment with different temperature values can help solidify their understanding of how temperature differentials drive efficiency in thermal systems.