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Chapter 3: Problem 32

A mixture of methane and air is capable of being ignited only if the mole percent of methane is between 5\% and 15\%. A mixture containing 9.0 mole\% methane in air flowing at a rate of 7.00 \(\times 10^{2} \mathrm{kg} / \mathrm{h}\) is to be diluted with pure air to reduce the methane concentration to the lower flammability limit. Calculate the required flow rate of air in mol/h and the percent by mass of oxygen in the product gas. (Note: Air may be taken to consist of \(\left.21 \text { mole } \% \mathrm{O}_{2} \text { and } 79 \% \mathrm{N}_{2} \text { and to have an average molecular weight of } 29.0 .\right)\)

Short Answer

Expert verified
The required flow rate of air is approximately \( 4403340\, mol/hr \) and the percent by mass of oxygen in the product gas is \( 23.54 \% \).

Step by step solution

01

Calculation of Moles of Methane and Air

First, calculate the original flow rate in moles per hour using the molecular weight of air and given mass flow rate. From there, find the moles of methane and air using the molar percentage of methane. Let \( x \) be the moles of air. \n\n Using the equation: \n\n \( \frac{9.0}{100} = \frac{7.00 脳 10^2 kg/h}{x (29.0 g/mol) 脳 (10^3 g/kg)} \)\n\n solving for \( x \), is found to be \( 2.718 脳 10^6 mol/h \). \nNow find methane and air moles as: \nMethane moles = \( \frac{9}{100} \times 2.718 脳 10^6 = 244620 mol/h \) \nAir moles = \( 2.718 脳 10^6 - 244620 = 2473380 mol/h \).
02

Calculation of Mole Ratio for Desired Methane Concentration

Now we aim to calculate the additional moles of air required to dilute the methane mixture to its lower flammability limit (5%). \n\n This is achieved by setting up the equation based on mole fractions: \n\n \( \frac{methane\,moles}{methane\,moles + Air\,moles + additional\,air\,moles} = 0.05 \) \n\n Inserting the previously calculated values: \n\n \( \frac{244620}{244620 + 2473380 +x} = 0.05 \), \n\n Solving this gives the \( x \) value to be \( 4403340\, mol/hr \). The total air moles now are \( 2573000 \, mol/hr \).
03

Calculation of Oxygen Mass Percentage

Finally, calculate the mass percentage of Oxygen in the product gas. Since the air consists of 21 mole% of Oxygen, moles of Oxygen will be \( 0.21 \times 6879620 \, (total\, moles) = 1444714 \, mol\). The total weight is given by \( (0.21 脳 6879620 脳 32) + (0.79 脳 6879620 脳 28) = 196497680 \, g \), using the molecular weight of \( \mathrm{O}_{2} \,and\, \mathrm{N}_{2} \) respectively. Hence, the mass percentage of oxygen can now be calculated as \n\n \( \frac{1444714 脳 32}{196497680 } 脳 100 \% = 23.54 \% \).

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Key Concepts

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

Flammability Limits
Flammability limits define the concentration range of a flammable gas or vapor in air that can ignite. For a methane-air mixture, this range is between 5% and 15% by volume of methane. Outside of these limits, the mixture is either too lean or too rich to support combustion.

Understanding these limits is crucial for safe industrial operations. For instance, if the methane concentration falls below 5%, the mixture won't ignite. Likewise, if it exceeds 15%, there's also no risk of flammability. This happens because, in a too-rich mixture, insufficient oxygen is available to sustain combustion.
  • The lower flammability limit (LFL) indicates the lowest concentration of a gas that will sustain a flame.
  • The upper flammability limit (UFL) is the highest concentration that allows ignition.
Keeping flammable gas concentrations within these limits can prevent explosion hazards. Engineers use dilution or enrichment techniques to control gas mixtures effectively and safely.
Methane-Air Mixtures
Methane is a primary component of natural gas and is widely used in various industrial processes. When mixed with air, methane creates a combustible bond with oxygen. The process efficiency and safety depend on maintaining specific methane concentrations.

In our example, the ratio of methane to air is initially 9% methane. Since it's within the flammability limits, there's a potential fire hazard. To render the mixture safe, we dilute it with additional air to decrease the methane concentration to 5%, the lower flammability limit, thus making it non-flammable.
  • Methane-air mixes follow simple stoichiometric rules for combustion reactions.
  • Proper ventilation and monitoring help maintain safe concentration levels.
Understanding the behavior of methane in air helps design safer chemical processes and avoid explosive environments.
Oxygen Mass Percentage
Oxygen is a significant component of air, typically comprising about 21% by volume. Calculating its mass percentage in a gas mixture involves considering the mole fraction and molecular weights of the gases involved.

In the methane dilution problem, after adjusting methane to the lower flammability limit, we calculate the Oxygen mass percentage in the new mixture. Here's how:
  • The total moles of the new gas mixture include moles from both the original and added air.
  • Oxygen moles are derived from multiplying the total moles by 0.21 (since air is 21% oxygen).
We then use the molecular weight of oxygen and nitrogen to compute the total weight of the mixture. Finally, by finding the proportion of this weight that is oxygen, we determine its mass percentage. In our case, it's about 23.54%, emphasizing that more air increases the overall oxygen share.

These calculations are vital in ensuring that the products of chemical processes are predictable and safe.

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Most popular questions from this chapter

You purchase six oranges that weigh a total of \(2 \mathrm{Ib}_{\mathrm{f}}\) and 13 ounces. After cutting them open and squeezing all the juice your strength allows into a large measuring cup, you weigh the remaining pulp and orange peels. They weigh 1 Ib \(_{\mathrm{f}}\) and 12 ounces and the total volume of the juice is 1.75 cups. What is the specific gravity of orange juice? State any assumptions you make.

In April \(2010,\) the worst oil spill ever recorded occurred when an explosion and fire on the Deepwater Horizon offshore oil-drilling rig left 11 workers dead and began releasing oil into the Gulf of Mexico. One of the attempts to contain the spill involved pumping drilling mud into the well to balance the pressure of escaping oil against a column of fluid (the mud) having a density significantly higher than those of seawater and oil. In the following problems, you may assume that seawater has a specific gravity of 1.03 and that the subsea wellhead was 5053 ft below the surface of the Gulf. (a) Estimate the gauge pressure (psig) in the Gulf at a depth of \(5053 \mathrm{ft}\). (b) Measurements indicate that the pressure inside the wellhead is 4400 psig. Suppose a pipe between the surface of the Gulf and the wellhead is filled with drilling mud and balances that pressure. Estimate the specific gravity of the drilling mud. (c) The drilling mud is a stable slurry of seawater and barite (SG \(=4.37\) ). What is the mass fraction of barite in the slurry? (d) What would you expect to happen if the barite weight fraction were significantly less than that estimated in Part (c)? Explain your reasoning.

Coal being used in a power plant at a rate of \(8000 \mathrm{lb}_{\mathrm{m}} / \mathrm{min}\) has the following composition: $$\begin{array}{lc}\hline \text { Component } & \text { Weight } \% \text { (dry basis) } \\ \hline \text { Ash } & 7.2 \\\\\text { Sulfur } & 3.5 \\\\\text { Hydrogen } & 5.0 \\\\\text { Carbon } & 75.2 \\ \text { Nitrogen } & 1.6 \\\\\text { Oxygen } & 7.5 \\\\\hline\end{array}$$ In addition, there are \(4.58 \mathrm{lb}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\) per \(\mathrm{lb}_{\mathrm{m}}\) of coal. Determine the molar flow rate of each element in the coal (including water) other than ash.

The following data have been obtained for the effect of solvent composition on the solubility of a serine, an amino acid, at \(10.0^{\circ} \mathrm{C}\) : $$\begin{array}{|l|c|c|c|c|c|c|c|c|}\hline \text { Volume \% Methanol } & 0 & 10 & 20 & 30 & 40 & 60 & 80 & 100 \\\\\hline \text { Solubility (g/100 mL solvent) } & 22.72 & 18.98 & 11.58 & 6.415 & 4.205 & 1.805 & 0.85 & 0.65 \\\\\hline \text { Solution Density (g/mL) } & 1.00 & 0.98 & 0.97 & 0.95 & 0.94 & 0.91 & 0.88 & 0.79 \\\\\hline\end{array}$$ The data were obtained by mixing known volumes of methanol and water to obtain the desired solvent compositions, and then slowly adding measured amounts of serine to each mixture until no more would go into solution. The temperature was held constant at \(10.0^{\circ} \mathrm{C}\). (a) Derive an expression for solvent composition expressed as mass fraction of methanol, \(x\), as a function of volume fraction of methanol, \(f\) (b) Prepare a table of solubility of serine (g serine/g solution) versus mass fraction of methanol.

The liquid level in a tank is determined by measuring the pressure at the bottom of the tank. A calibration curve is prepared by filling the tank to several known levels, reading the bottom pressure from a Bourdon gauge, and drawing a plot of level (m) vs. pressure (Pa). (a) Would you expect the calibration curve to be a straight line? Explain your answer. (b) The calibration experiment was done using a liquid with a specific gravity of \(0.900,\) but the tank is used to store a liquid with specific gravity of \(0.800 .\) Will the liquid level determined from the calibration curve be too high, too low, or correct? Explain. (c) If the actual liquid level is 8.0 meters, what value will be read from the calibration curve? If the tank has a height of \(10.0 \mathrm{m},\) what value will be read from the curve when the tank overflows?
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