91Ó°ÊÓ

n-Pentane is burned with excess air in a continuous combustion chamber. (a) A technician runs an analysis and reports that the product gas contains 0.270 mole\% pentane, \(5.3 \%\) oxygen, \(9.1 \%\) carbon dioxide, and the balance nitrogen on \(a\) dry basis. Assume 100 mol of dry product gas as a basis of calculation, draw and label a flowchart, perform a degree-offreedom analysis based on atomic species balances, and show that the system has -1 degree of freedom. Interpret this result. (b) Use balances to prove that the reported percentages could not possibly be correct. (c) The technician reruns the analysis and reports new values of 0.304 mole\% pentane, \(5.9 \%\) oxygen, \(10.2 \%\) carbon dioxide, and the balance nitrogen. Verify that this result could be correct and, assuming that it is, calculate the percent excess air fed to the reactor and the fractional conversion of pentane. (d) It was emphasized in Part (c) that the new composition could be correct. Explain why it isn't possible to say for sure; illustrate your response by considering a set of equations with -1 degree of freedom.

Step by step solution

01

Calculating Degree of Freedom

A degree of freedom analysis involves counting the number of unknowns versus the number of independent equations you have to solve for them. The stoichiometry of the combustion reaction of pentane (C5H12) is as follows \(C5H12 + 8O2 → 5CO2 + 6H2O\) . Hence, the number of unknowns in this case are the moles of pentane, oxygen, carbon dioxide, and water vapor, which gives us a total of 4 unknowns. Since we are dealing with an excess air scenario, the moles of nitrogen are not unknown. As for the number of independent equations at hand, they are obtained from the atomic species balances (i.e., carbon balance, hydrogen balance, and oxygen balance). There are 3 such independent equations. Therefore, the degree of freedom is calculated as the number of unknowns minus the number of independent equations, which in this case equals 1 (4-3). But the question mentions the degree of freedom to be -1, which is a discrepancy.

02

Validity of Reported Percentages

In order to check if the reported percentages of product gases are plausible, insert them into the species balance equations that were derived from reacting stoichiometry. If the results do not conform to said equations, then these percentages are incorrect.

03

Verification of Revised Percentages and Calculation of Excess Air and Conversion

Perform a similar process for the revised values. If these new compositions match the species balance equations, then they are correct. Calculate the amount of excess air in the feed by comparing the moles of oxygen present in the products to the amount required for complete combustion of pentane. Furthermore, the fractional conversion of pentane is calculated by comparing the initial number of moles of pentane with the final number of unreacted moles (if any) in the products.

04

Reasoning on Certainty

When there is -1 degree of freedom, this indicates there is an additional variable that is currently not being considered. One cannot definitively assert the correctness of the compositions without further information.

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Key Concepts

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

Combustion Reaction Stoichiometry

Understanding the stoichiometry of a combustion reaction is critical when analyzing chemical processes involving the burning of a compound. Stoichiometry involves the quantitative relationship between the reactants and products in a chemical reaction. In the context of combustion, this typically involves a hydrocarbon (like n-pentane) reacting with oxygen to produce carbon dioxide, water, and other gases.

For instance, the stoichiometric combustion of n-pentane (\(C_5H_{12}\)) can be represented by the equation: \[C_5H_{12} + 8O_2 \rightarrow 5CO_2 + 6H_2O\].

This means that one mole of n-pentane requires eight moles of oxygen to completely react to form five moles of carbon dioxide and six moles of water. In real-world scenarios, however, the reaction would typically take place with excess air to ensure complete combustion. This is important for analyses because it affects the composition of the product gases and their subsequent use in calculations such as species balance equations and determining the percentage of excess air.

Degree of Freedom Analysis

A degree of freedom analysis is an essential part of chemical process analysis. It helps determine if the system of equations describing the process has a unique solution. The degrees of freedom are calculated as the number of independent variables minus the number of independent equations. In chemical processes, the independent variables usually include the moles or concentrations of reactants and products, while the independent equations arise from mass balances, energy balances, and species balances.

In our example, with the combustion of n-pentane, we consider four unknowns (moles of pentane, oxygen, carbon dioxide, and water vapor) and three atomic species balances (carbon, hydrogen, and oxygen), resulting in one degree of freedom. However, negative degrees of freedom, as suggested by the exercise, indicate an over-specified system and typically point to an error in the data or the assumption of excess species that was not accounted for. This discrepancy requires a re-evaluation of the system's parameters.

Species Balance Equations

Species balance equations are foundational to solving combustion and chemical reaction problems. They elucidate how different elements and compounds are transformed and conserved in the reaction process. The key to species balances is the law of conservation of mass, which states that in a closed system not subjected to a nuclear reaction, the mass of an element must remain constant over the course of a chemical reaction.

In the given problem, we derive species balances for carbon, hydrogen, and oxygen. For example, the carbon balance ensures that the moles of carbon in n-pentane entering the reaction are equal to the moles of carbon in the carbon dioxide and any unreacted pentane in the output. \[C_5H_{12} \rightarrow xCO_2 + yC_5H_{12}\]
The reported percentages can be converted to mole percentages and checked against these balance equations to see if the reported outcome is reasonable. This not only allows us to verify the accuracy of the technician's report but also helps us understand the conversion efficiency and the excess air provided during the combustion process.