Question

When a man returns to his well-sealed house on a summer day, he finds that the house is at \(35^{\circ} \mathrm{C}\). He turns on the air conditioner, which cools the entire house to \(20^{\circ} \mathrm{C}\) in 30 min. If the \(\mathrm{COP}\) of the air-conditioning system is \(2.8,\) determine the power drawn by the air conditioner. Assume the entire mass within the house is equivalent to \(800 \mathrm{kg}\) of air for which \(c_{v}=0.72 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\) and \(c_{p}=1.0 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\)

Short answer

Answer: The power drawn by the air conditioner is 8,572 Watts (8.572 kW).

Step-by-step solution

Calculate the heat extracted from the air.

First, we need to find the amount of heat extracted by the air conditioner from the air inside the house. We can use the formula for heat absorbed or released by a substance: \(Q = m × C × \Delta T\) where \(Q\) is the heat extracted, \(m\) is the mass of air, \(C\) is the heat capacity at constant pressure for air (\(c_p\)), and \(\Delta T\) is the temperature difference. Given values: mass \(m = 800\) kg heat capacity \(c_p = 1.0\ \mathrm{kJ/kg\cdot ^\circ C}\) \(\Delta T = 35^\circ\mathrm{C} - 20^\circ\mathrm{C} = 15^\circ\mathrm{C}\) Replacing the values, we have: \(Q = 800 \times 1.0 \times 15\)

Calculate the heat extracted.

Now, let's calculate the heat extracted: \(Q = 800 \times 1.0 \times 15 = 12000\ \mathrm{kJ}\) The air conditioner extracted 12,000 kJ of heat from the air inside the house.

Use the COP value to find the power drawn by the air conditioner.

Now, we will use the coefficient of performance (COP) to find the power drawn. The formula for COP is: \(\mathrm{COP} = \dfrac{Q_\text{Extracted}}{W_\text{Drawn}}\) where \(Q_\text{Extracted}\) is the heat extracted from the air (in kJ) and \(W_\text{Drawn}\) is the power drawn by the air conditioner (in kW). Given COP \(= 2.8\), we can rearrange the formula for the power drawn (\(W_\text{Drawn}\)): \(W_\text{Drawn} = \dfrac{Q_\text{Extracted}}{\mathrm{COP}}\) Replace the values: \(W_\text{Drawn} = \dfrac{12000}{2.8}\)

Calculate the power drawn by the air conditioner.

Let's now calculate the power drawn by the air conditioner: \(W_\text{Drawn} = \dfrac{12000}{2.8} = 4286 \ \mathrm{kW}\) Now we have calculated the power drawn by the air conditioner, but it's important to consider that it cools the house in 30 minutes, so we need to divide the result by the time in hours (0.5). \(W_\text{Drawn} = \dfrac{4286}{0.5} = 8572 \ \mathrm{W}\) So, the power drawn by the air conditioner is 8,572 Watts (8.572 kW) as it cools the house to 20°C in 30 minutes.