Question

A \(3-m \times 4-m \times 5-m\) room is to be heated by one ton \((1000 \mathrm{kg})\) of liquid water contained in a tank that is placed in the room. The room is losing heat to the outside at an average rate of \(6000 \mathrm{kJ} / \mathrm{h}\). The room is initially at \(20^{\circ} \mathrm{C}\) and 100 \(\mathrm{kPa}\) and is maintained at an average temperature of \(20^{\circ} \mathrm{C}\) at all times. If the hot water is to meet the heating requirements of this room for a 24 -h period, determine the minimum temperature of the water when it is first brought into the room. Assume constant specific heats for both air and water at room temperature.

Short answer

Answer: The minimum initial temperature of the water should be approximately 54.38°C.

Step-by-step solution

Calculate the heat loss during 24 hours.

Since the room is losing heat at a rate of 6000 kJ/h, we'll first find the total amount of heat loss during 24 hours. Total_heat_loss = Heat_loss_rate x Time Total_heat_loss = 6000 kJ/h x 24 h Total_heat_loss = 144000 kJ

Calculate the heat required from the water.

Since we know the total heat loss, we now need to calculate the heat required from the one-ton water tank to maintain the room temperature at 20°C. First, we need to determine the difference between the initial and final temperatures of the water; let's denote the minimum initial temperature of the water as T_initial. heat_transfer_amount = Total_heat_loss = m_water * c_water * (T_initial - T_final) Where: m_water = 1000 kg (mass of the water) c_water = 4.186 kJ/kg°C (specific heat capacity of water at room temperature) T_final = 20 °C (the final temperature, which is the room temperature)

Solve for the initial temperature of water.

Now we can solve for the initial temperature of water by re-arranging the formula derived in Step 2. T_initial - T_final = Total_heat_loss / (m_water * c_water) T_initial - 20 °C = 144000 kJ / (1000 kg * 4.186 kJ/kg°C) T_initial - 20 °C = 34.38 °C T_initial = 34.38 °C + 20 °C T_initial = 54.38 °C Thus, the minimum initial temperature of the water when it is first brought into the room should be approximately 54.38°C.