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An oxygen tank with a volume of \(2.5 \mathrm{ft}^{3}\) is kept in a room at \(50^{\circ} \mathrm{F}\). An engineer has used the idealgas equation of state to determine that if the tank is first evacuated and then charged with \(35.3 \mathrm{lb}_{\mathrm{m}}\) of pure oxygen, its rated maximum allowable working pressure (MAWP) will be attained. Operation at pressures above this value is considered unsafe. (a) What is the maximum allowable working pressure (psig) of the tank? (b) You suspect that at the conditions of the fully charged tank, ideal-gas behavior may not be a good assumption. Use the SRK equation of state to obtain a better estimate of the maximum mass of oxygen that may be charged into the tank. Did the ideal-gas assumption lead to a conservative estimate (on the safe side) or a nonconservative estimate of the amount of oxygen that could be charged? (c) Suppose the tank is charged and ruptures before the amount of oxygen calculated in Part (b) enters it. (It should have been able to withstand pressures up to four times the MAWP.) Think of at least five possible explanations for the failure of the tank below its rated pressure limit.

Step by step solution

01

Calculating Maximum Allowable Working Pressure鈥�

Since we know the volume of the tank, the temperature, and the mass of oxygen, we can use the ideal gas law to solve for the pressure. The ideal gas law, usually written as PV=nRT, allows us to estimate the behaviour of gases under most conditions. Now, use the unit conversions, and the molecular weight of oxygen to convert the mass of oxygen into the amount of substance n (given in moles). After determining the number of moles, plug the given numbers with the gas constant for the ideal gas law equation to estimate the pressure.

02

Estimating Maximum Mass of Oxygen Using the SRK Equation

To confirm whether the assumption of the ideal gas law was correct, we must test it against another equation of state - the SRK equation. The SRK equation accommodates interactions between gas particles and provides a more accurate estimate in this case. After determining SRK constants for oxygen, plug these constants and given conditions into the SRK equation to calculate the most accurate estimate for the pressure. This pressure and the SRK equation are then used to estimate the maximum mass of oxygen.

03

Checking the Ideal Gas Law Assumption

Compare the mass of oxygen charge obtained from the SRK equation to that obtained from the ideal gas law. If the mass obtained from the ideal gas law is smaller, then the assumption was conservative (safe). If the mass obtained from the ideal gas law is larger, the assumption was nonconservative.

04

Understanding Possible Reasons for Tank Failure

There could be several reasons why the tank fails to hold the amount of oxygen predicted. Some possible reasons may include defective manufacturing, damage to the tank, change in environmental conditions, physical overfilling, or even operator error.

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Key Concepts

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

Oxygen Tank Pressure

When dealing with gases stored in a tank, particularly in safety-sensitive environments like oxygen storage, precision in calculating pressure is crucial. The pressure in an oxygen tank can be calculated using equations like the ideal gas law, expressed as \(PV = nRT\). Here, \(P\) is the pressure, \(V\) is the volume, \(n\) is the amount in moles of the gas, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin. For the oxygen tank, this equation provides an estimate for maximum allowable working pressure (MAWP) based on the condition of the gas.

In practice, if the estimated pressure targets exceed the tank's rated MAWP, the setup will be marked as unsafe for operation. This ensures that the tank can withstand the internal gas pressure without rupturing. The correct estimation of pressure is not just about accuracy, but directly ties into safety and usability of the equipment.

SRK Equation of State

While the ideal gas law offers a simplified model for understanding gas behavior, it can't always account for all real-world interactions, especially at high pressures or low temperatures. This is where the SRK (Soave-Redlich-Kwong) equation of state comes in. It improves upon the ideal gas law by incorporating factors for intermolecular forces and the specific volume occupied by gas molecules.

The SRK equation is expressed as:\[P = \frac{RT}{V-b} - \frac{a\alpha}{(V+b)V}\]where \(a\) and \(b\) are substance-specific constants, and \(\alpha\) considers temperature effects. This calculation provides a better estimate for systems involving high pressures or non-ideal gas behaviors.

In the context of an oxygen tank, using the SRK equation can result in a different number of moles of gas that can be safely stored under the given conditions. This precise calculation is essential for ensuring safety standards are met.

Gas Behavior Assumptions

Different models are used to assume the behavior of gases in various conditions. The ideal gas law assumes a kind of perfect world where gas particles do not interact and occupy no space, which is rarely the case in reality. In our oxygen tank example, using this model might give a conservative estimate since it might not fully reflect potential increases in pressure caused by intermolecular forces.

Conversely, the SRK equation of state provides a more nuanced picture by taking real-world interactions into account. It allows for more accurate evaluations, especially for conditions close to or beyond the critical points of gases. Understanding these assumptions is critical, as it informs the design and operation decisions, impacting both performance and safety.

Tank Safety and Failure Analysis

Analyzing why a tank might fail involves looking at numerous potential causes. Even if calculations predict safety under specified conditions, unexpected factors can lead to a tank failure. Some common explanations include:

• Defective Manufacturing: Material flaws or poor workmanship can make tanks unable to withstand calculated pressures.
• Physical Damage: Past mishandling or unnoticed injuries to the tank can compromise its structural integrity.
• Environmental Changes: Variations in external pressure or temperature can inadvertently raise tank pressure.
• Overfilling: Incorrect charging amounts beyond calculated values can put undue stress on the tank walls.
• Operator Errors: Misjudgments in operation or maintenance can also lead to premature failures.

By closely examining these factors, engineers can develop better protocols and designs to mitigate risks, enhancing the overall reliability of the storage system.