Question

Consider a large classroom with 90 students on a hot summer day. All the lights with \(2.0 \mathrm{~kW}\) of rated power are kept on. The room has no external walls, and thus heat gain through the walls and the roof is negligible. Chilled air is available at \(15^{\circ} \mathrm{C}\), and the temperature of the return air is not to exceed \(25^{\circ} \mathrm{C}\). The average rate of metabolic heat generation by a person sitting or doing light work is \(115 \mathrm{~W}\) ( \(70 \mathrm{~W}\) sensible and \(45 \mathrm{~W}\) latent). Determine the required flow rate of air that needs to be supplied to the room.

Short answer

Answer: The required flow rate of air to maintain the maximum temperature of 25°C in the classroom is approximately 1.03 m³/s.

Step-by-step solution

Calculate the total sensible heat

Calculate the total sensible heat generated in the room considering the heat from the students and the lights. Sensible heat generated by students = (90 students * 70 W) = 6300 W. Heat generated by the lights = 2000 W. So, the total sensible heat generated in the room = 6300 W (students) + 2000 W (lights) = 8300 W.

Calculate the total latent heat

Calculate the total latent heat generated in the room by considering the heat from the students: Latent heat generated by the students = 90 students * 45 W = 4050 W.

Calculate the heat gain ratio

Determine the heat gain ratio of sensible to latent heat by dividing the total sensible heat by the total latent heat. Heat gain ratio = Total Sensible Heat / Total Latent Heat = 8300 W (sensible) / 4050 W (latent) ≈ 2.05.

Calculate the required cooling capacity

Determine the total heat that the cooling system must be able to remove to maintain the maximum return air temperature. Cooling capacity = Total Sensible Heat + Total Latent Heat = 8300 W (sensible) + 4050 W (latent) = 12350 W.

Calculate the temperature difference between the supply and return air

Assuming the air is cooled completely to 15°C before being supplied and has to be maintained at 25°C upon return, find the difference in temperatures. ∆T = 25°C (return air temperature) – 15°C (supply air temperature) = 10°C.

Determine the specific heat of air

Assume the specific heat of air (Cp) is about 1005 J/kg·K.

Calculate the mass flow rate of air

Determine the mass flow rate of air (m) required to balance the heat gains in the room by using the formula: m = Cooling Capacity / (Cp * ∆T). m = 12350 W / (1005 J/kg·K * 10°C) ≈ 1.23 kg/s.

Convert mass flow rate to volume flow rate

Using the formula: volume flow rate (Q) = mass flow rate (m) / air density (ρ). Assuming the air density (ρ) is 1.2 kg/m³, calculate the volume flow rate. Q = 1.23 kg/s / 1.2 kg/m³ ≈ 1.03 m³/s. Therefore, the required flow rate of air to be supplied to the room is approximately 1.03 m³/s.

Key concepts

Sensible Heat Calculation

Sensible heat refers to the energy needed to change the temperature of air without changing its phase. In our classroom scenario, the main sources of sensible heat come from the body's metabolic activities and electrical equipment like lights.

To determine the total sensible heat in the room, we add up the heat generated by each source:

• Students: With 90 students producing 70 W of sensible heat each, the total heat contributed by students is 6300 W.
• Lights: The room's lights add an additional 2000 W to the total heat load.

By combining these figures, we find the room has a total sensible heat load of 8300 W. This value is crucial for determining how much cooling is required.

Latent Heat Calculation

Latent heat represents the energy necessary to change the phase of air moisture content, contributing to humidity control without affecting temperature directly. In this case, the students inside the classroom are the primary contributors of latent heat.

Each student emits 45 W of latent heat, resulting in a total latent heat contribution from all students calculated as follows:

• Latent heat from students: 90 students × 45 W = 4050 W

This value helps in assessing the overall cooling demand since both sensible and latent heats contribute to the total heat gain.

Air Flow Rate Calculation

Calculating the air flow rate is essential to determine how much air needs to be circulated to maintain desired indoor conditions. It's done to ensure that the cooling supplied by the air matches the total heat gains in the room.

Once the cooling capacity necessary to remove both the sensible (8300 W) and latent heat (4050 W) is known, you follow these steps:

• Temperature Difference (ΔT): The difference between the supply air (15°C) and the return air (25°C) is 10°C.
• Specific Heat of Air (Cp): Generally assumed to be 1005 J/kg·K.

Using the formula, the mass flow rate is calculated:
\[ m = \frac{12350 \, \text{W}}{1005 \, \text{J/kg·K} \times 10 \, ^\circ\text{C}} \approx 1.23 \, \text{kg/s} \]

Cooling Capacity

The cooling capacity signifies the total ability of a cooling system to remove heat from a designated area. It is the combination of both sensible and latent heat values required to maintain a comfortable environment.

In our classroom example, the cooling capacity necessary is the sum of all heat contributions:

• Sensible Heat: 8300 W
• Latent Heat: 4050 W

So, the total cooling capacity needed is 8300 W + 4050 W = 12350 W. Ensuring that cooling equipment can meet this capacity is vital for effective climate control.

Specific Heat of Air

The specific heat of air, denoted as \(Cp\), is the amount of energy required to raise the temperature of a unit mass of air by one degree Celsius (or one Kelvin). For typical calculations, the specific heat of air is assumed to be about 1005 J/kg·K.

This property is integral to thermal calculations because it links temperature changes to energy consumption. In our situation, \(Cp\) helps us find the mass flow rate, ensuring the cooling system can adequately adjust the air temperature.