Question

A thermoelectric generator receives heat from a source at \(340^{\circ} \mathrm{F}\) and rejects the waste heat to the environment at \(90^{\circ} \mathrm{F}\). What is the maximum thermal efficiency this thermoelectric generator can have?

Short answer

Answer: To calculate the maximum thermal efficiency, use the Carnot efficiency formula: efficiency = \(1 - \frac{T_{2K}}{T_{1K}}\), where \(T_{1K}\) is the heat source temperature in Kelvin, and \(T_{2K}\) is the environment temperature in Kelvin. First, convert the temperatures to Kelvin: \(T_{1K} = (340 - 32) \times \frac{5}{9} + 273.15\) \(T_{2K} = (90 - 32) \times \frac{5}{9} + 273.15\) Next, substitute the values and solve for the efficiency: Efficiency = \(1 - \frac{(90 - 32) \times \frac{5}{9} + 273.15}{(340 - 32) \times \frac{5}{9} + 273.15}\) Calculate the efficiency and express the result as a percentage to find the maximum thermal efficiency of the thermoelectric generator.

Step-by-step solution

Convert temperatures to Kelvin

To calculate the Carnot efficiency, we need the temperatures in Kelvin. We can convert Fahrenheit to Kelvin using the following formula: \(T_{K} = (T_{F} - 32) \times \frac{5}{9} + 273.15\). Heat source temperature in Fahrenheit T1 = \(340^{\circ} \mathrm{F}\) Environment temperature in Fahrenheit T2 = \(90^{\circ} \mathrm{F}\) Convert both to Kelvin: \(T_{1K} = (340 - 32) \times \frac{5}{9} + 273.15\) \(T_{2K} = (90 - 32) \times \frac{5}{9} + 273.15\)

Calculate Carnot efficiency

Now, we can calculate the maximum thermal efficiency using the Carnot efficiency formula, which is: efficiency = \(1 - \frac{T_{2K}}{T_{1K}}\), where \(T_{1K}\) is the heat source temperature in Kelvin and \(T_{2K}\) is the environment temperature in Kelvin. Efficiency = \(1 - \frac{T_{2K}}{T_{1K}}\)

Substitute the values and solve

Substitute the values of \(T_{1K}\) and \(T_{2K}\) that we calculated in step 1, and solve for the maximum thermal efficiency. Efficiency = \(1 - \frac{(90 - 32) \times \frac{5}{9} + 273.15}{(340 - 32) \times \frac{5}{9} + 273.15}\) Calculate the efficiency and express the result as a percentage.

Interpret the result

The resulting value represents the maximum thermal efficiency that the thermoelectric generator can achieve. This means that it is the highest possible percentage of heat energy from the heat source that can be converted into useful work, while the remaining energy will be rejected to the environment as waste heat.