Question

The temperature of the air in a building can be maintained at a desirable level during winter by using different methods of heating. Compare heating this air in a heat exchanger unit with condensing steam to heating it with an electric-resistance heater. Perform a second-law analysis to determine the heating method that generates the least entropy and thus causes the least exergy destruction.

Short answer

Answer: The electric-resistance heater is considered the most efficient method for heating air in a building during winter, as it generates less entropy (-2424.06 J/K) compared to the heat exchanger unit with condensing steam (-6016.35 J/K).

Step-by-step solution

Calculate the entropy generation for the heat exchanger unit with condensing steam

We know that the entropy generation for a heat transfer process can be calculated using the formula: \(S_{gen} = Q \times \frac{1}{T_H} - Q \times \frac{1}{T_L}\) where \(Q\) is the heat transferred, \(T_H\) is the temperature of the high-temperature reservoir (the hot steam in this case), and \(T_L\) is the temperature of the low-temperature reservoir (the colder air that we want to heat). For simplicity, let's assume that the heat exchanged is 1 kWh (3600000 J), which should be enough to heat a medium-sized room. We need to know the temperatures of the hot steam and the colder air inside the building as Kelvin values to perform the calculation. Let's assume the required air temperature in the building is 20°C (293 K) and the steam temperature is 100°C (373 K). Now we can calculate the entropy generation: \(S_{gen, steam} = 3600000 \times \frac{1}{373} - 3600000 \times \frac{1}{293} = -6016.35 J/K\)

Calculate the entropy generation for the electric-resistance heater

For the electric-resistance heater, the entropy generation corresponds to the energy dissipation to the ambient environment. The entropy generation in this case can be calculated using the same formula: \(S_{gen} = Q \times \frac{1}{T_H} - Q \times \frac{1}{T_L}\) where \(Q\) is the heat transferred, \(T_H\) is the temperature of the high-temperature reservoir (the electric resistance heater in this case), and \(T_L\) is the temperature of the low-temperature reservoir (the colder air that we want to heat). Let's assume the electric resistance heater has a temperature of 80°C (353 K). Now we can calculate the entropy generation: \(S_{gen, electric} = 3600000 \times \frac{1}{353} - 3600000 \times \frac{1}{293} = -2424.06 J/K\)

Compare the entropy generation results

Now that we have calculated the entropy generation for both the heat exchanger unit with condensing steam and the electric-resistance heater, we can compare the results: \(S_{gen, steam} = -6016.35 J/K\) \(S_{gen, electric} = -2424.06 J/K\) A lower entropy generation corresponds to less exergy destruction, which means the process is more efficient. In this case, the electric-resistance heater generates less entropy and should be considered the most efficient method for heating the air in the building.