Question

A well-sealed room contains \(60 \mathrm{kg}\) of air at \(200 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C}\). Now solar energy enters the room at an average rate of \(0.8 \mathrm{kJ} / \mathrm{s}\) while a \(120-\mathrm{W}\) fan is turned on to circulate the air in the room. If heat transfer through the walls is negligible, the air temperature in the room in 30 min will be \((a) 25.6^{\circ} \mathrm{C}\) (b) \(49.8^{\circ} \mathrm{C}\) \((c) 53.4^{\circ} \mathrm{C}\) \((d) 52.5^{\circ} \mathrm{C}\) \((e) 63.4^{\circ} \mathrm{C}\)

Short answer

Answer: Final Temperature = Initial Temperature + (Total Energy / (Mass of Air * Specific Heat Capacity))

Step-by-step solution

Determine the total energy entering the room

First, we need to find the total energy entering the room due to the fan and solar energy. Since the fan has a power of 120 W, it is converting its energy at rate of 120 J/s. The solar energy is entering the room at 0.8 kJ/s, which is equal to 800 J/s. To find the total energy entering the room in 30 minutes, we convert the time to seconds, and then multiply that by the rate of energy entering. Total Time (s) = 30 minutes * 60 s/minute = 1800 s Total energy (J) = (Fan power (J/s) + Solar power (J/s)) * Total Time (s) Total energy (J) = (120 J/s + 800 J/s) * 1800 s

Calculate the temperature rise

Next, we will use the specific heat capacity of air and the total energy entering the room to calculate the temperature rise. The specific heat capacity of air at constant pressure is approximately 1005 J/kgK. Temperature Rise = Total Energy / (Mass of Air * Specific Heat Capacity) Temperature Rise = Total Energy / (60 kg * 1005 J/kgK)

Determine the final temperature

To find the final temperature in the room after 30 minutes, add the temperature rise to the initial temperature of 25°C. Final Temperature = Initial Temperature + Temperature Rise Now we can plug the expressions from steps 1 and 2 to find the final temperature and choose the correct option from the given choices.