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A gas leak has led to the presence of 1.00 mole \(\%\) carbon monoxide in a \(350-\mathrm{m}^{3}\) laboratory. \(^{4}\) The leak was discovered and sealed, and the laboratory is to be purged with clean air to a point at which the air contains less than the OSHA (Occupational Safety and Health Administration) specified Permissible Exposure Level (PEL) of 35 ppm (molar basis). Assume that the clean air and the air in the laboratory are atthe same temperature and pressure and that the laboratory air is perfectly mixed throughout the purging process. (a) Let \(t_{\mathrm{r}}(\mathrm{h})\) be the time required for the specified reduction in the carbon monoxide concentration. Write a differential CO mole balance, letting \(N\) equal the total moles of gas in the room (assume constant), the mole fraction of CO in the room air, and \(\dot{V}_{\mathrm{p}}\left(\mathrm{m}^{3} / \mathrm{h}\right)\) the flow rate of purge air entering the room (and also the flow rate of laboratory air leaving the room). Convert the balance into an equation for \(d x / d t\) and provide an initial condition. Sketch a plot of \(x\) versus \(t,\) labeling the value of \(x\) at \(t=0\) and the asymptotic value at \(t \rightarrow \infty\) (b) Integrate the balance to derive an equation for \(t_{r}\) in terms of \(\dot{V}_{\mathrm{p}}\) (c) If the volumetric flow rate is \(700 \mathrm{m}^{3} / \mathrm{h}\) (representing a tumover of two room volumes per hour), how long will the purge take? What would the volumetric flow rate have to be to cut the purge time in half? (d) Give several reasons why it might not be safe to resume work in the laboratory after the calculated purge time has elapsed. What precautionary steps would you advise taking at this point?

Step by step solution

01

Derive a Differential Mole Balance

The required mole balance is derived by taking into account that the change in the number of moles of CO in the room over time is equal to the rate of CO entering minus the rate of CO leaving. This can be written as \(\frac{dN_{CO}}{dt} = \dot{V}_{p}x_{0} - \dot{V}_{p}x\), where\(N_{CO}\) are the moles of CO in the room at time t, \(x_{0}\) is the initial mole fraction of CO, and \(x\) is the mole fraction of CO at time t. Since the total moles of gas in the room is constant, we can write\(N_{CO}= Nx\) with N the total moles of gas, leading to \(N\frac{dx}{dt} = \dot{V}_{p}(x_{0}-x)\). The initial condition is \(x(0) = x_{0}\). For the plot, \(x\) will decrease gradually from \(x_{0}\) at \(t=0\) to an asymptotic value of 0 as \(t \rightarrow \infty\).

02

Integration of the Balance Equation

To find the time needed to reduce CO to acceptable limits, integrate the differential equation. Separation of variables and subsequent integration results in \(N\int_{x_{0}}^{x}\frac{dx}{x_{0}-x}= \dot{V}_{p}\int_{0}^{t}dt\), which simplifies to \(-N\log\frac{x}{x_{0}}= \dot{V}_{p} t\). Solving for \(t\) provides an expression for the necessary time for the CO concentration to be reduced from \(x_{0}\) to \(x\), namely \(t=-\frac{N}{ \dot{V}_{p}}\log\frac{x}{x_{0}}\).

03

Time for Purification and Required Flow Rate

The permissible exposure level is 35 ppm or 35x10^-6 in mole fraction. Substituting this for \(x\), the initial \(x_{0}\) of 1 mole%, or 0.01 in mole fraction, the total moles of gas \(N\) as volume/22.4 since 1 mole occupies 22.4 m^3 at standard conditions, and \(\dot{V}_{p}\) of 700 m^3/h, we find \(t=-\frac{350/22.4}{700}\log\frac{35x10^-6}{0.01} \approx 3.99\) hours. To cut the purge time in half, we need to double the purge rate to 1400 m^3/h.

04

Safety Precautions

Resuming work in the lab immediately after the calculated purge time might not be safe because the calculation assumes perfect mixing of the gases, which may not be the case. It also assumes that the CO source has been completely stopped, but there may still be residual CO. As a precaution, the CO level should be monitored before resuming operations, the area should be properly ventilated, and an inspection of the lab should be performed to identify any potential sources of residual CO.

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Key Concepts

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

Gas Purging

Gas purging is a vital safety procedure that ensures harmful gases, such as carbon monoxide, are removed from an enclosed space. The process involves flushing the enclosed area with a clean, inert gas—often air—to reduce the concentration of unwanted gases to a safe level. In the context of industrial settings or laboratories, achieving a low concentration is crucial to prevent health hazards.
When conducting gas purging, it is important to:

• Calculate the volume of the space accurately to determine the right flow rate for the purging gas.
• Use an appropriate flow rate to ensure that the gas mixture is effectively diluted and replaced over time.
• Maintain uniform conditions such as temperature and pressure to enhance mixing throughout the area.

Effective gas purging requires a proper understanding of gas dynamics and the specific characteristics of the space being purged. Continuous monitoring of gas levels ensures that the process is successful and that the air quality is restored to a safe standard.

Carbon Monoxide Exposure

Carbon monoxide (CO) is a colorless, odorless, and toxic gas that poses significant health risks upon exposure. Even at low concentrations, CO can lead to serious health effects, which makes understanding its exposure levels critical, especially in occupational environments. The Occupational Safety and Health Administration (OSHA) has established a permissible exposure limit (PEL) for workplaces, which is 35 parts per million (ppm). Exceeding this limit can result in symptoms ranging from headaches and dizziness to much more severe health complications, like impaired cognitive function, or even death in extreme cases.
To protect against CO exposure:

• Regular monitoring of air quality is important, to quickly identify any rise in CO levels above safe limits.
• Install CO detectors in strategic locations, especially in areas prone to gas accumulation.
• Ensure adequate ventilation and continue to educate workers about recognizing potential symptoms of CO exposure.

Understanding the properties of CO and maintaining vigilant safety measures creates a safer and healthier environment.

Differential Equations

Differential equations are mathematical equations that describe how quantities change over time, often used in modeling dynamic systems in various fields including engineering and physics. In the case of gas purging, they help describe how the concentration of a gas changes as clean air is introduced.
To model the reduction of carbon monoxide concentration, a differential equation balances the input and output rates of the gases involved. For example, the rate of CO leaving the laboratory is balanced against the mole fraction of CO present and the volumetric flow rate of the incoming clean air:\[\frac{dx}{dt} = -\frac{\dot{V}_{p}}{N}(x - x_{0})\]This equation simplifies understanding the relationship between different variables involved in gas purging. Solving this differential equation helps to determine the time required to achieve a desired concentration level, ensuring that safety standards are met. Understanding and correctly applying differential equations is essential for accurate modeling and decision-making in engineering contexts.

Occupational Safety

Occupational safety is crucial in any workplace to protect workers from health hazards that may occur due to exposure to harmful substances or unsafe practices. In scenarios involving gas leaks, such as the carbon monoxide instance in the exercise, it is paramount to have effective safety protocols in place.
Key considerations for occupational safety in such situations include:

• Thoroughly checking that leaks are sealed before attempting to purify the air.
• Ensuring that the purging process is complete and has achieved concentrations below the acceptable limits before resuming work.
• Performing regular safety audits and training sessions to keep workers informed and ready to act during emergencies.

Resuming activities without ensuring the safe completion of these steps might still leave workers exposed to potential health risks. Investing in preventive measures, ongoing monitoring, and upkeep of equipment reduces risks and creates a safer working environment. Understanding and implementing occupational safety standards keeps both workers and businesses protected.