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Chapter 5: Problem 12

In a workshop with a capacity of \(10^{4} \mathrm{~m}^{3}\) fans deliver \(10^{3} \mathrm{~m}^{3}\) of fresh air per minute, containing \(0.04 \% \mathrm{CO}_{2}\), and the same amount of air is vented to the outside. At 9:00 AM the workers arrive and after half an hour, the content of \(\mathrm{CO}_{2}\) in the air rises to \(0.12 \%\). Evaluate the carbon dioxide content of the air by 2:00 PM.

Short Answer

Expert verified
Answer: The carbon dioxide content at 2:00 PM will be less than 0.12%.

Step by step solution

01

Calculate the total air exchanged every minute

We are given that the fans deliver \(10^{3}\mathrm{~m}^{3}\) of fresh air per minute and the same amount of air is vented to the outside. So, the total air exchanged every minute is \(2\times10^{3} \mathrm{~m}^{3}\).
02

Determine the total hours worked and amount of CO鈧� inhaled

The workers arrived at 9:00 AM, and by 2:00 PM, they worked for 5 hours. After half an hour, they inhaled enough CO鈧� to increase the content to 0.12%. We need to find how much CO鈧� was inhaled during that time. The workshop capacity is \(10^{4} \mathrm{~m}^{3}\), and CO鈧� increased by \(0.08\%(0.12\%-0.04\%)\). So, the amount of CO鈧� inhaled is \((10^{4}\mathrm{~m}^{3})(0.08\%)=8\mathrm{~m}^{3}\).
03

Calculate the rate of CO鈧� inhalation per hour

To find the rate of CO鈧� inhalation per hour, divide the amount inhaled by the time it took to inhale that amount. The time taken was 0.5 hours, and the amount inhaled was 8 m鲁. So, the rate of CO鈧� inhalation is \(\frac{8\mathrm{~m}^{3}}{0.5 \mathrm{~h}}=16\mathrm{~m}^{3/h}\).
04

Calculate the CO鈧� inhaled from 9:30 AM to 2:00 PM

Now, we will find the amount of CO鈧� inhaled during the 4.5 hours from 9:30 AM to 2:00 PM. As the rate of CO鈧� inhalation is 16 m鲁/h, the total inhaled CO鈧� during that time period is \((16\mathrm{~m}^{3/h})(4.5\mathrm{~h})=72\mathrm{~m}^{3}\).
05

Find the total CO鈧� vented from 9:30 AM to 2:00 PM

We are given that for every minute, \(10^{3}\mathrm{~m}^{3}\) of fresh air enters, and the same amount of air is vented to the outside. Thus, in 4.5 hours, or 270 minutes, a total of \((10^{3}\mathrm{~m}^{3/min})(270\mathrm{~min})= 270\times10^{3}\mathrm{~m}^{3}\) of mixed air has been vented outside. On average, this mixed air had a CO鈧� content of 0.08%. So, the total vented CO鈧� is \((270\times10^{3}\mathrm{~m}^{3})(0.08\%)=216\mathrm{~m}^{3}\).
06

Calculate the CO鈧� content at 2:00 PM

Now, we need to find the total amount of CO鈧� in the workshop at 2:00 PM. The initial CO鈧� content was 0.12% of \(10^{4} \mathrm{~m}^{3}\), which is \(12\mathrm{~m}^{3}\). Then, from 9:30 AM to 2:00 PM, 72 m鲁 CO鈧� was inhaled, and 216 m鲁 CO鈧� was vented. So, the total CO鈧� content in the workshop at 2:00 PM will be: \(12\mathrm{~m}^{3}+72\mathrm{~m}^{3}-216\mathrm{~m}^{3}=-132\mathrm{~m}^{3}\). As the resultant is negative, it means there's a reduction in the CO鈧� inside the workshop. So, by 2:00 PM, the atmosphere in the workshop has less CO鈧� than when the workers started. So, the carbon dioxide content at 2:00 PM will be less than 0.12%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mathematical Analysis
The process of calculating the carbon dioxide content in a given volume of air, particularly in an enclosed environment such as a workshop, revolves around the mathematical analysis. This involves logical reasoning and the application of mathematical concepts to quantify the amount of CO2 based on given conditions such as ventilation rates and time.

For instance, in our example, calculating the total air exchanged every minute requires basic multiplication, while determining the rate of CO2 inhalation per hour involves division. Both calculations serve as foundational steps for assessing the CO2 accumulation over time. To ensure clarity, it's essential to follow these steps sequentially, which builds upon the previous information, eventually leading to the final calculation of CO2 content at a specific time.
CO2 Concentration Problem
The CO2 concentration problem focuses on assessing the amount of CO2 present in an environment over time, given the activity within the space, such as human occupancy, and the effectiveness of the ventilation system at removing or introducing CO2. The central challenge is to quantify this concentration at a particular moment, which in our example is 2:00 PM.

In such problems, it's crucial to take into account the baseline concentration of CO2, any sources that could increase CO2 (like human respiration), and the systems in place that regulate its levels through ventilation. The mathematical analysis relies heavily on these variables, showing the importance of accounting for all factors that influence the given environment's CO2 concentration.
Ventilation Rates
Ventilation rates are a key factor when analyzing the air quality within indoor spaces. They dictate the volume of air that is replaced within a certain period and therefore play a crucial role in any mathematical model assessing the concentration of pollutants like CO2.

In terms of improving the calculation process for students, it鈥檚 important to highlight that understanding ventilation rates allows us to model how pollutants are diluted or accumulate over time. For our CO2 concentration problem, the rate at which fresh air is introduced and contaminated air is vented directly influences the CO2 content. To solve problems related to air quality, students should grasp how to factor in the ventilation rates efficiently, which is crucial for accurate environmental health assessments.
Exponential Decay
Exponential decay is a mathematical concept that describes the process of a quantity decreasing over time at a rate proportional to its current value. While it's commonly associated with radioactive decay, it's also highly relevant in the study of air pollutants, as contaminants can decrease in concentration exponentially due to effective ventilation.

However, the present example does not involve an exponential decay in the strictest sense because a constant volume of air鈥攁nd, thus, CO2鈥攍eaves the room instead of a proportionate amount. This points to the importance of context when applying mathematical models. Even so, understanding exponential decay enriches a student's problem-solving toolkit for various situations, including those involving gradually decreasing or increasing quantities over time.

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Most popular questions from this chapter

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