Question

Consider an office room that is being cooled adequately by a 12,000 Btu/h window air conditioner. Now it is decided to convert this room into a computer room by installing several computers, terminals, and printers with a total rated power of \(8.4 \mathrm{kW}\). The facility has several \(7000 \mathrm{Btu} / \mathrm{h}\) air conditioners in storage that can be installed to meet the additional cooling requirements. Assuming a usage factor of 0.4 (i.e., only 40 percent of the rated power will be consumed at any given time) and additional occupancy of seven people, each generating heat at a rate of \(100 \mathrm{W}\), determine how many of these air conditioners need to be installed to the room.

Short answer

Answer: To find out how many air conditioners need to be installed, follow these steps: 1. Calculate the heat generated by the computers, terminals, and printers: 0.4 * 8400 W = 3360 W 2. Calculate the heat generated by the occupants: 7 * 100 W = 700 W 3. Calculate the total heat generated in the room: 3360 W + 700 W = 4060 W 4. Convert the total heat generated to Btu/h: 4060 W * 3.412 Btu/h = 13844.32 Btu/h 5. Calculate the additional cooling requirement: 13844.32 Btu/h - 12,000 Btu/h = 1844.32 Btu/h 6. Determine how many air conditioners are needed: 1844.32 Btu/h ÷ 7000 Btu/h = 0.263 ≈ 1 So, 1 additional 7000 Btu/h air conditioner needs to be installed to meet the cooling requirements of the room.

Step-by-step solution

Calculate heat generated by the computers, terminals, and printers

First, we need to calculate the total heat generated by the computers, terminals, and printers. The usage factor is given as 0.4 (40% of the rated power will be consumed). So, the total heat generated by the equipment is: \(Q_{equipment} = 0.4 \times 8.4 \mathrm{kW}\) Convert the power to watts: \(Q_{equipment} = 0.4 \times 8400 \mathrm{W}\)

Calculate heat generated by occupants

There are 7 people in the room, each generating heat at a rate of 100 W. The total heat generated by the occupants is: \(Q_{occupants} = 7 \times 100 \mathrm{W}\)

Calculate total heat generated

Now, we'll add the heat generated by the equipment and the occupants to find the total heat generated in the room: \(Q_{total} = Q_{equipment} + Q_{occupants}\)

Calculate additional cooling requirement

As 12,000 Btu/h window air conditioner is already installed, we need to calculate the additional cooling requirement. First, convert the total heat generated (in watts) to Btu/h by using the conversion factor: 1 W = 3.412 Btu/h. \(Q_{total} = Q_{total} \times 3.412 \mathrm{Btu/ h}\) Now, subtract the cooling capacity of the installed air conditioner to find the additional cooling required: \(Q_{additional} = Q_{total} - 12,000 \mathrm{Btu/h}\)

Calculate how many air conditioners need to be installed

Now that we know how much additional cooling is required, we can determine how many \(7000 \mathrm{Btu} / \mathrm{h}\) air conditioners are needed. To do this, divide the additional cooling requirement by the capacity of the available air conditioners: \(N = \frac{Q_{additional}} {7000 \mathrm{Btu/h}}\) Since we cannot install a fraction of an air conditioner, round up to the nearest whole number – this will provide the minimum number of air conditioners needed to adequately cool the room.

Key concepts

Heat Generation Calculation

Understanding the heat generation in an environment is crucial for determining the cooling requirements of a space. In the context of the exercise provided, heat generation primarily comes from two sources: electronic equipment and human occupancy.

For electronic equipment, the heat output can often be approximated based on the power rating of the devices and a usage factor. This factor represents the proportion of time the devices are expected to operate at full power. The calculation is fairly straightforward: first, multiply the total rated power of the equipment by the usage factor to get the power consumption in kilowatts (kW), then convert kW to watts (W) by multiplying by 1,000.

When dealing with human occupancy, it's common to estimate that each person will contribute a certain amount of heat, often around 100 W, into the room. To calculate the heat generation from occupancy, simply multiply the number of occupants by the average heat output per person.

In the provided example, we consider both the heat from the electronics and the human occupants to calculate the total heat generation in the room.

Cooling Load Analysis

A cooling load analysis involves evaluating all sources of heat within a space to determine how much cooling is needed to maintain a comfortable temperature.

In our example, this entails combining the heat generated by both the electronic equipment and the people inside the room. To determine a room's cooling load, we must calculate the heat produced by all sources and convert this value into a common unit of measurement, such as British Thermal Units per hour (BTU/h), which is widely used in heating and cooling applications.

The cooling load analysis will take into account the existing air conditioner's capacity and the additional heat load to find out how much more cooling is needed. This analysis is pivotal for selecting the right size and number of air conditioning units to ensure the environment remains at the desired temperature.

BTU per Hour Conversion

Once the total heat generation within a room is calculated in watts, it needs to be converted into British Thermal Units per hour (BTU/h) to match the common output measurements of air conditioning systems.

The conversion rate from watts to BTU/h is a critical figure: 1 W equals 3.412 BTU/h. This means you multiply the power in watts by 3.412 to get the equivalent power in BTU/h. This step is essential to ensure the compatibility between the heat generated in the space and the cooling capacity of the air conditioning units, as different countries and industries may use different units to measure these values.

In the exercise, after calculating the equipment and occupancy heat generation in watts, they are then converted into BTU/h, allowing us to directly compare against the capacity of air conditioning units measured in the same units.

Occupancy Heat Contribution

People naturally emit heat, and when we have several individuals in a confined space, such as an office or computer room, this heat contribution can significantly affect the temperature of the room.

In the context of heating, ventilation, and air conditioning (HVAC) design and analysis, this metabolic heat release is considered an internal heat gain. To estimate the heat gain from occupancy, a general assumption is that each person will give off approximately 100 W of heat. By calculating the heat contribution for each occupant, we achieve a more accurate assessment of the cooling needs of the space.

For example, with seven people in the room, the total heat output from occupants alone is 700 W. This heat must be included in the total cooling requirement calculation to ensure that the selected air conditioning units can handle the actual heat load and maintain comfortable room conditions.