Question

An auditorium is \(100 \mathrm{~m}\) in length, \(70 \mathrm{~m}\) in width, and \(20 \mathrm{~m}\) in height. It is ventilated by a system that feeds in fresh air and draws out air at the same rate. Assume that airborne impurities form a well-stirred mixture. The ventilation system is required to reduce air pollutants present at any instant to \(1 \%\) of their original concentration in \(30 \mathrm{~min}\). What inflow (and outflow) rate is required? What fraction of the total auditorium air volume must be vented per minute?

Short answer

Answer: The required inflow/outflow rate is approximately 46.20 m^3/min, and the fraction of the total auditorium air volume that must be vented per minute is approximately 0.00033 or 0.033%.

Step-by-step solution

Calculate the volume of the auditorium

The volume of a rectangular room or box can be calculated as: Volume = length × width × height Given that length = 100 m, width = 70 m, and height = 20 m, the volume of the auditorium is: Volume = 100 m × 70 m × 20 m = 140,000 m^3.

Set up the equation with the given conditions

We need to reduce the pollutant concentration to 1% of its original amount in 30 minutes. Let V be the total volume of the auditorium, R be the inflow/outflow rate, and t be the time in minutes. The following equation holds: (V - R × t) × 0.01 = R × t

Substitute the values into the equation

We know the volume V = 140,000 m^3, and time t = 30 minutes. Substitute these values into the equation: (140,000 - R × 30) × 0.01 = R × 30

Solve the equation for R

Now, solve the equation for R: 1,400 - 0.3 × R = R × 30 1,400 = 30.3 × R Divide both sides of the equation by 30.3 to find the inflow/outflow rate, R: R = \frac{1,400}{30.3} ≈ 46.20 \mathrm{~m^3/min}

Calculate the fraction of the total auditorium air volume vented per minute

The required inflow/outflow rate is 46.20 m^3/min. To find the fraction of the total volume of the auditorium that is vented per minute, divide the inflow/outflow rate by the total volume of the auditorium: Fraction = \frac{46.20}{140,000} ≈ 0.00033 The required inflow (and outflow) rate is approximately 46.20 m^3/min, and the fraction of the total auditorium air volume that must be vented per minute is approximately 0.00033 or 0.033%.

Key concepts

Differential Equations in Environmental Engineering

Differential equations serve as a mathematical foundation in environmental engineering for modeling various phenomena, including the dynamics of pollutants in the air. A typical use case involves describing how the concentration of an airborne contaminant changes over time within a given volume due to ventilation.

To achieve this, engineers set up differential equations that reflect the balance between the rate of contaminant removal through the ventilation system and the rate of accumulation or decay of the contaminant within the environment. In the case of the auditorium problem, we implicitly used a differential equation by setting up an equilibrium condition corresponding to the desired reduction in pollutant concentration over time.

Air Quality Control

Air quality control is essential in creating healthy indoor environments. Engineers apply various strategies to manage air quality, such as filtering, purifying, and ventilating. Effective ventilation is crucial as it dilutes and removes pollutants. In our exercise, the ventilation system's role is to maintain air quality by reaching a specific target concentration of pollutants.

Regulations might dictate these target levels, and engineers need to design systems capable of achieving them within a set time frame. Regular maintenance and monitoring are also part of effective air quality control to ensure systems operate correctly and efficiently.

Exponential Decay Model

The exponential decay model is commonly used to describe how certain quantities decrease over time at a rate proportional to their value. This model applies perfectly to the ventilation scenario where contaminants in the air reduce exponentially as fresh air replaces polluted air.

In our auditorium example, the aim is for pollutants to reach 1% of their original concentration in 30 minutes. This objective indicates an exponential decay process because the reduction is a constant percentage of the current concentration, characteristic of such models. Understanding the nature of exponential decay helps in designing systems to control environmental parameters effectively.

Ventilation Rate Computation

Calculating the ventilation rate is an essential task in environmental engineering for ensuring adequate air quality. The ventilation rate is the speed at which fresh air replaces indoor air within a certain volume, like the auditorium in our exercise.

The computation involves establishing a relationship between the volume of the space, the desired decrease in pollutants, and the time frame for achieving this. By applying the exponential decay model, we figure out the necessary ventilation rate to meet the requirements. The fraction of air replaced per unit time, obtained from our calculations, allows engineers to specify the performance required of ventilation equipment.