Problem 44 - Precalculus Student Solutions Manual 5th

Question

Classroom Ventilation According to the American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE), a nonsmoking classroom should have a ventilation rate of \(15 \mathrm{ft}^{3}\) per min for each person in the room. (a) Write an equation that models the total ventilation \(V\) (in cubic feet per hour) necessary for a classroom with \(x\) students. (b) A common unit of ventilation is air change per hour (ach). 1 ach is equivalent to exchanging all of the air in a room every hour. If \(x\) students are in a classroom having volume \(15,000 \mathrm{ft}^{3},\) determine how many air exchanges per hour \((A)\) are necessary to keep the room properly ventilated. (c) Find the necessary number of ach ( \(A\) ) if the classroom has 40 students in it. (d) In areas like bars and lounges that allow smoking, the ventilation rate should be increased to \(50 \mathrm{ft}^{3}\) per min per person. Compared to classrooms, ventilation should be increased by what factor in heavy smoking areas?

Step-by-Step Solution

Answer

a) V = 900x. b) A = \frac{x}{16.67}. c) A ≈ 2.4 ach for 40 students. d) The ventilation should be increased by approximately 3.33 times in smoking areas.

Step 1 - Model the Total Ventilation Rate

Begin by noting the ASHRAE recommendation of 15 cubic feet per minute (ft³/min) per person. For a classroom with x students, the total ventilation rate required per minute is: \[ V_{minute} = 15x \] To convert this to cubic feet per hour (ft³/hr), multiply by 60 (since there are 60 minutes in an hour): \[ V_{hour} = 15x \times 60 = 900x \]

Step 2 - Convert Ventilation Rate to Air Changes per Hour (ach)

Given the classroom volume is 15,000 ft³, calculate the air changes per hour (ach) by dividing the hourly ventilation rate by the classroom volume: \[ A = \frac{V_{hour}}{15,000} = \frac{900x}{15,000} = \frac{3x}{50} = \frac{x}{16.67} \]

Step 3 - Calculate ach for 40 students

Substitute x = 40 into the equation for ach: \[ A = \frac{40}{16.67} \ \approx 2.4 \]

Step 4 - Compare Ventilation Requirements for Smoking Areas

The ventilation rate for smoking areas is 50 ft³/min per person. To find the factor of increase: \[ \text{Factor} = \frac{50}{15} = \frac{10}{3} \approx 3.33 \]

Key Concepts

Ventilation Rate

Ventilation rate refers to the amount of fresh air that needs to be supplied to a space per unit of time. In our exercise, we focus on the classroom setting. According to ASHRAE recommendations, a classroom should be ventilated at a rate of 15 cubic feet per minute (ft³/min) for each student. This ensures a constant supply of fresh air, keeping the indoor air quality good for everyone. For a classroom with x students, the total ventilation rate per minute can be calculated using the formula: \( V_{minute} = 15x \). To convert this to an hourly rate, we multiply by 60 (the number of minutes in an hour), resulting in \( V_{hour} = 900x \).

Air Changes per Hour (ach)

Air Changes per Hour, or ach, is a standard unit that tells us how many times the air within a room is completely replaced within one hour. It is essential for assessing the effectiveness of a ventilation system. For a classroom with volume 15,000 ft³ and x students, the hourly ventilation rate is \( 900x \). The number of air changes per hour can be calculated by dividing the hourly ventilation rate by the room volume: \( A = \frac{900x}{15,000} \). Simplifying this, we get \( A = \frac{x}{16.67} \). This result tells us the number of times the air is exchanged in the classroom every hour.

ASHRAE Recommendations

The American Society of Heating, Refrigerating, and Air-Conditioning Engineers (ASHRAE) provides guidelines to ensure indoor air quality and comfort. For classrooms, ASHRAE recommends a ventilation rate of 15 cubic feet per minute per person. This ensures that the air remains fresh and helps to remove contaminants and CO₂. ASHRAE guidelines change based on the type of space; for example, spaces that permit smoking have a higher recommended ventilation rate due to the additional pollutants introduced by smoking.

Total Ventilation

Total ventilation refers to the overall amount of air that a space needs in order to maintain good air quality. For our classroom example, the total ventilation required in cubic feet per hour is evaluated for x students using the equation: \( V_{hour} = 900x \). By considering the number of students in the given space, we can make sure that the ventilation system is appropriately sized. If there are 40 students in a classroom, the necessary total ventilation in cubic feet per hour would be: \( V_{hour} = 900 \times 40 = 36,000 \) cubic feet per hour.

Classroom Volume

Classroom volume is the total space within the classroom measured in cubic feet. It is a critical factor in determining how effective a ventilation system will be. For example, in our problem, the classroom volume is given as 15,000 ft³. Knowing the classroom volume helps calculate the number of air changes per hour, a key measure of how air quality is maintained. The volume provides a basis for determining whether the calculated ventilation rate is sufficient to maintain good air quality by ensuring the air within the classroom is replaced multiple times per hour according to the recommended standards.