Question

A classroom that normally contains 40 people is to be air-conditioned with window air-conditioning units of \(5-\mathrm{kW}\) cooling capacity. A person at rest may be assumed to dissipate heat at a rate of about \(360 \mathrm{kJ} / \mathrm{h}\). There are 10 lightbulbs in the room, each with a rating of \(100 \mathrm{W}\). The rate of heat transfer to the classroom through the walls and the windows is estimated to be \(15,000 \mathrm{kJ} / \mathrm{h}\). If the room air is to be maintained at a constant temperature of \(21^{\circ} \mathrm{C},\) determine the number of window air-conditioning units required.

Short answer

Answer: 2 window air-conditioning units are required.

Step-by-step solution

Calculate heat generated by people

Each person at rest dissipates heat at a rate of 360 kJ/h. We have 40 people in the classroom, so the total heat generated by people is: Total heat generated by people = Number of people * Heat generated by each person Total heat generated by people = 40 * 360 kJ/h Total heat generated by people = 14,400 kJ/h

Calculate heat generated by lightbulbs

There are 10 lightbulbs, each rated at 100 W. We need to convert this value to kJ/h (since the other rates are in kJ/h) as follows: 1 W = 1 J/s 100 W * (3600 s/h) = 360 kJ/h So, each lightbulb generates 360 kJ/h. Therefore, the total heat generated by all lightbulbs is: Total heat generated by lightbulbs = Number of lightbulbs * Heat generated by each lightbulb Total heat generated by lightbulbs = 10 * 360 kJ/h Total heat generated by lightbulbs = 3,600 kJ/h

Calculate the total heat generated in the room

Now, we will add the heat generated by people, lightbulbs, and the heat transfer through walls and windows to find the total heat generated in the room. Total heat generated = Heat generated by people + Heat generated by lightbulbs + Heat transfer through walls and windows Total heat generated = 14,400 kJ/h + 3,600 kJ/h + 15,000 kJ/h Total heat generated = 33,000 kJ/h

Determine the cooling capacity needed

To maintain the room at a constant temperature of 21°C, the cooling capacity of the air-conditioning units should be equal to the total heat generated in the room. We already have the total heat generated in kJ/h, so: Cooling capacity needed = Total heat generated Cooling capacity needed = 33,000 kJ/h Now, we need to convert the cooling capacity from kJ/h to kW: 1 kW = 3600 kJ/h Cooling capacity needed = 33,000 kJ/h * (1 kW / 3600 kJ/h) Cooling capacity needed ≈ 9.167 kW

Calculate the number of air conditioning units required

Each window air-conditioning unit has a cooling capacity of 5 kW. To find the number of air conditioning units required, we will divide the cooling capacity needed by the cooling capacity of each unit: Number of air conditioning units = Cooling capacity needed / Cooling capacity of each unit Number of air conditioning units = 9.167 kW / 5 kW Number of air conditioning units ≈ 1.833 Since we cannot have a fraction of an air conditioning unit, we will round up to the nearest whole number to ensure that the total cooling capacity is sufficient. Number of air conditioning units = 2 Therefore, 2 window air-conditioning units are required to maintain the room at a constant temperature of 21°C.

Key concepts

Cooling Capacity

Understanding cooling capacity is essential when considering the requirements for air conditioning a space. It is the measure of an air conditioning unit's ability to remove heat from an area and is usually expressed in kilowatts (kW). To put it simply, the higher the cooling capacity, the more heat that unit can remove from the room per hour. In our classroom example, each air conditioning unit has a cooling capacity of 5 kW, which means it can remove 5 kilowatts of heat per hour.

When calculating the total cooling capacity needed, it's important to consider all heat sources, including human occupants, electronics, and heat transfer through the building's envelope. By summing up all heat contributions and converting them to a common unit (in this case, kJ/h, then to kW), we can determine how much cooling capacity is required to maintain a constant desirable temperature inside the classroom.

Heat Generation by Occupants

The presence of people in a confined space significantly contributes to the overall heat level due to their body heat. In our example, heat generation by occupants is estimated based on a rate at which a person at rest dissipates heat—about 360 kJ per hour. With 40 occupants in the classroom, this collectively adds up to 14,400 kJ/h.

This aspect of heat gain is often overlooked but is vital in calculations for air conditioning needs. The total heat generated by occupants can vary depending on their activity level, number, and even clothing. This needs to be taken into consideration alongside other heat sources to determine the accurate cooling capacity required for the space.

Air Conditioning Units

The role of air conditioning units is to maintain a comfortable environment by removing the excess heat from a space. They come in various sizes and capacities to suit different needs. Our exercise illustrates the use of window air-conditioning units, which are commonly used for single rooms or small spaces.

When choosing air conditioning units, it's important to match their total cooling capacity with the heat load of the room. In this case, the calculated total cooling capacity needed is approximately 9.167 kW, indicating that at least two units with a capacity of 5 kW each are necessary to maintain the desired temperature of 21°C. This ensures that the cumulative cooling capacity exceeds the heat gain, accounting for any potential variations in heat generation throughout the day.