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Chapter 4: Problem 84

A gas contains 75.0 wt\% methane, \(10.0 \%\) ethane, \(5.0 \%\) ethylene, and the balance water. (a) Calculate the molar composition of this gas on both a wet and a dry basis and the ratio (mol \(\mathrm{H}_{2} \mathrm{O} /\) mol dry gas). (b) If \(100 \mathrm{kg} / \mathrm{h}\) of this fuel is to be burned with \(30 \%\) excess air, what is the required air feed rate (kmol/ h)? How would the answer change if the combustion were only \(75 \%\) complete?

Short Answer

Expert verified
The molar composition of the gas on a wet and a dry basis can be found using the weight percentages of the gases. The molar ratio of water to the dry gas is also calculated. The air feed rate and its requirement for complete and incomplete combustion are calculated using stoichiometric balance for the combustion reaction of each gas. A reduction in air feed rate is seen for incomplete combustion.

Step by step solution

01

Calculate molar composition on a wet basis

Given the weight percent of methane, ethane and ethylene, first convert these into mole fractions. Use the molar mass for each gas and calculate the total number of moles. The molar composition of each component can be calculated by dividing the number of specific moles by the total moles.
02

Calculate molar composition on a dry basis

For the dry basis calculation, the mole fraction of water vapor has to be excluded. The new total moles would be sum of moles of methane, ethane and ethylene only and mole fraction can be calculated similarly as in step 1.
03

Calculating ratio of water to dry gas

This ratio is essentially the mole fraction of water in the gas.
04

Calculate air feed rate for complete combustion

Use the stoichiometric balance for the combustion reaction of each gas to find out the moles of oxygen and hence of air required. The total air feed rate is the sum of air needed for combustion of all the gases. The excess air needs to be accounted for.
05

Calculate the change in air feed rate for incomplete combustion

If the combustion is not complete, the moles of air required would be less. We need to calculate the air feed rate for 75% completion of combustion by multiplying the initial air feed rate by 0.75.

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Key Concepts

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

Wet Basis Molar Composition
The molar composition on a wet basis refers to the percentage of each component in a gas mixture, including the water vapor. To calculate this composition, start by using the weight percentages of the components provided in the exercise. Convert these weight percentages into mole fractions by using the molar masses of each gas component, such as methane, ethane, ethylene, and water.

Once you have the mole fractions, calculate the total number of moles by adding up all the moles of each component. To find the molar composition of each component, divide the moles of a specific gas by the total moles. This gives you the composition on a wet basis, which includes all gas components present in the mixture.
Dry Basis Molar Composition
When we talk about dry basis molar composition, we're referring to the molar composition of the gas mixture without considering the water vapor. To determine this composition, start by removing the moles of water vapor from the total moles calculated in the wet basis method.

With the water moles removed, sum up the moles of the remaining gases: methane, ethane, and ethylene. Calculate the mole fraction for each of these gases by dividing the moles of each gas by the new total number of moles (without water).

This new set of mole fractions represents the molar composition on a dry basis. This perspective helps in processes where the water content in the gas can affect the outcome, such as combustion calculations.
Combustion Air Feed Rate
To ensure proper combustion of the gas mixture, you need to calculate the air feed rate, which is the amount of air required for the chemical reaction. This requires using stoichiometric balance principles to determine the exact amount of oxygen needed.

Combustion requires oxygen, and since air is about 21% oxygen, you'll calculate the moles of oxygen required and then convert it into moles of air. Don鈥檛 forget to factor in excess air, as mentioned, 30% more than the stoichiometric amount.

This ensures complete combustion. If 100 kg/h of fuel is burned with this air rate, calculate in kmol/h by using the moles of oxygen needed.
Stoichiometric Balance
Stoichiometric balance is crucial in combustion calculations. It describes the exact proportions of reactants and products in a chemical reaction. In combustion, this involves balancing the chemical equation to ensure all methane, ethane, and ethylene react with the right amount of oxygen.

Using stoichiometric coefficients from the balanced equations, calculate the moles of oxygen needed for combustion. This balance helps determine the theoretical air feed rate.

If combustion is incomplete, adjust calculations to reflect the 75% completion by taking 75% of the calculated air feed rate. This alteration showcases the influence of chemical reaction efficiency on required reactant quantities.

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Most popular questions from this chapter

Ethylene oxide is produced by the catalytic oxidation of ethylene: $$ 2 \mathrm{C}_{2} \mathrm{H}_{4}+\mathrm{O}_{2} \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O} $$ An undesired competing reaction is the combustion of ethylene: $$ \mathrm{C}_{2} \mathrm{H}_{4}+3 \mathrm{O}_{2} \longrightarrow 2 \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O} $$ The feed to the reactor (not the fresh feed to the process) contains 3 moles of ethylene per mole of oxygen. The single-pass conversion of ethylene is \(20 \%,\) and for every 100 moles of ethylene consumed in the reactor, 90 moles of ethylene oxide emerge in the reactor products. A multiple-unit process is used to separate the products: ethylene and oxygen are recycled to the reactor, ethylene oxide is sold as a product, and carbon dioxide and water are discarded. (a) Assume a quantity of the reactor feed stream as a basis of calculation, draw and label the flowchart, perform a degree-of-freedom analysis, and write the equations you would use to calculate (i) the molar flow rates of ethylene and oxygen in the fresh feed, (ii) the production rate of ethylene oxide, and (iii) the overall conversion of ethylene. Do no calculations. (b) Calculate the quantities specified in Part (a), either manually or with an equation-solving program. (c) Calculate the molar flow rates of ethylene and oxygen in the fresh feed needed to produce 1 ton per hour of ethylene oxide.

A variation of the indicator-dilution method (see preceding problem) is used to measure total blood volume. A known amount of a tracer is injected into the bloodstream and disperses uniformly throughout the circulatory system. A blood sample is then withdrawn, the tracer concentration in the sample is measured, and the measured concentration [which equals (tracer injected)/(total blood volume) if no tracer is lost through blood vessel walls] is used to determine the total blood volume. In one such experiment, \(0.60 \mathrm{cm}^{3}\) of a solution containing \(5.00 \mathrm{mg} / \mathrm{L}\) of a dye is injected into an artery of a grown man. About 10 minutes later, after the tracer has had time to distribute itself uniformly throughout the bloodstream, a blood sample is withdrawn and placed in the sample chamber of a spectrophotometer. A beam of light passes through the chamber, and the spectrophotometer measures the intensity of the transmitted beam and displays the value of the solution absorbance (a quantity that increases with the amount of light absorbed by the sample). The value displayed is 0.18. A calibration curve of absorbance \(A\) versus tracer concentration \(C\) (micrograms dye/liter blood) is a straight line through the origin and the point \((A=0.9, C=3 \mu \mathrm{g} / \mathrm{L}) .\) Estimate the patient's total blood volume from these data.

The gas-phase reaction between methanol and acetic acid to form methyl acetate and water takes place in a batch reactor. When the reaction mixture comes to equilibrium, the mole fractions of the four reactive species are related by the reaction equilibrium constant $$K_{y}=\frac{y_{C} y_{D}}{y_{A} y_{B}}=4.87$$ (a) Suppose the feed to the reactor consists of \(n_{\mathrm{A} 0}, n_{\mathrm{B} 0}, n_{\mathrm{C} 0}, n_{\mathrm{D} 0},\) and \(n_{10}\) gram-moles of \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},\) and an inert gas, I, respectively. Let \(\xi\) be the extent of reaction. Write expressions for the gram-moles of each reactive species in the final product, \(n_{\mathrm{A}}(\xi), n_{\mathrm{B}}(\xi), n_{\mathrm{C}}(\xi),\) and \(n_{\mathrm{D}}(\xi) .\) Then use these expressions and the given equilibrium constant to derive an equation for \(\xi_{c}\), the equilibrium extent of reaction, in terms of \(\left.n_{\mathrm{A} 0}, \ldots, n_{10} . \text { (see Example } 4.6-2 .\right)\) (b) If the feed to the reactor contains equimolar quantities of methanol and acetic acid and no other species, calculate the equilibrium fractional conversion. (c) It is desired to produce 70 mol of methyl acetate starting with 75 mol of methanol. If the reaction proceeds to equilibrium, how much acetic acid must be fed? What is the composition of the final product? (d) Suppose it is important to reduce the concentration of methanol by making its conversion at equilibrium as high as possible, say 99\%. Again assuming the feed to the reactor contains only methanol and acetic acid and that it is desired to produce 70 mol of methyl acetate, determine the extent of reaction and quantities of methanol and acetic acid that must be fed to the reactor. (e) If you wanted to carry out the process of Part (b) or (c) commercially, what would you need to know besides the equilibrium composition to determine whether the process would be profitable? (List several things.)

A mixture of propane and butane is burned with pure oxygen. The combustion products contain 47.4 mole \(\% \mathrm{H}_{2} \mathrm{O}\). After all the water is removed from the products, the residual gas contains 69.4 mole \(\% \mathrm{CO}_{2}\) and the balance \(\mathrm{O}_{2}\) (a) What is the mole percent of propane in the fuel? (b) It now turns out that the fuel mixture may contain not only propane and butane but also other hydrocarbons. All that is certain is that there is no oxygen in the fuel. Use atomic balances to calculate the elemental molar composition of the fuel from the given combustion product analysis (i.e., what mole percent is \(C\) and what percent is \(\mathrm{H}\) ). Prove that your solution is consistent with the result of Part (a).

Fermentation of sugars obtained from hydrolysis of starch or cellulosic biomass is an alternative to using petrochemicals as the feedstock in production of ethanol. One of the many commercial processes to do this \(^{16}\) uses an enzyme to hydrolyze starch in corn to maltose (a disaccharide consisting of two glucose units) and oligomers consisting of several glucose units. A yeast culture then converts the maltose to ethyl alcohol and carbon dioxide: $$\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}+\mathrm{H}_{2} \mathrm{O}(+\text { yeast }) \rightarrow 4 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}+4 \mathrm{CO}_{2}\left(+\text { yeast }+\mathrm{H}_{2} \mathrm{O}\right)$$ As the yeast grows, \(0.0794 \mathrm{kg}\) of yeast is produced for every \(\mathrm{kg}\) ethyl alcohol formed, and \(0.291 \mathrm{kg}\) water is produced for every kg of yeast formed. For use as a fuel, the product from such a process must be around 99.5 wt\% ethyl alcohol. Corn fed to the process is 72.0 wt\% starch on a moisture-free basis and contains 15.5 wt\% moisture. It is estimated that 101.2 bushels of corn can be harvested from an acre of com, that each bushel is equivalent to \(25.4 \mathrm{lb}_{\mathrm{m}}\) of corn, and that \(6.7 \mathrm{kg}\) of ethanol can be obtained from a bushel of corn. What acreage of farmland is required to produce 100,000 kg of ethanol product? What factors (economic and environmental) must be considered in comparing production of ethanol by this route with other routes involving petrochemical feedstocks?
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