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Chapter 4: Problem 18

One thousand kilograms per hour of a mixture containing equal parts by mass of methanol and water is distilled. Product streams leave the top and the bottom of the distillation column. The flow rate of the bottom stream is measured and found to be \(673 \mathrm{kg} / \mathrm{h}\), and the overhead stream is analyzed and found to contain 96.0 wt\% methanol. (a) Draw and label a flowchart of the process and do the degree-of-freedom analysis. (b) Calculate the mass and mole fractions of methanol and the molar flow rates of methanol and water in the bottom product stream. (c) Suppose the bottom product stream is analyzed and the mole fraction of methanol is found to be significantly higher than the value calculated in Part (b). List as many possible reasons for the discrepancy as you can think of. Include in your list possible violations of assumptions made in Part (b).

Short Answer

Expert verified
The flowchart represents the distillation column with the input, top product and bottom product streams. The mass fractions of methanol and water in the bottom product stream are determined using flow rate and composition information, and these are then converted to mole fractions. The molar flow rates of methanol and water are calculated by dividing their mass flow rates by their respective molar masses. Potential causes for discrepancy between calculated and actual mole fractions could be measurement error, volatile column conditions, violation of ideal behavior assumptions, and variability in control of the input flow rate.

Step by step solution

01

Drawing and Labeling of a Flowchart

Begin with drawing a flowchart to better visualize the process. The input stream should be represented by a single arrow going into the distillation column. Two arrows should then emanate from the column 鈥� one leading from the top (for the overhead stream) and one leading from the bottom (for the bottom product stream). Label these arrows with their respective flows and compositions. Also perform the degree-of-freedom analysis, which involves identifying the unknowns and equations in the system, and ensuring that the number of equations is equal to the number of unknowns.
02

Calculating Mass and Mole Fractions

Next, the mass fractions of methanol and water in the bottom product stream need to be determined. Since the total input flow rate is known (1000 kg/h), as is the flow rate of the bottom product stream (673 kg/h), the flow rate of the overhead stream can be calculated as the difference between the two. Knowing the composition of this overhead stream, the amount of methanol in it can be calculated. Subtracting this from the original amount of methanol in the input stream gives the amount of methanol in the bottom product stream. Dividing this by the total mass of the bottom product stream gives the mass fraction of methanol. The mass fraction of water can be found in a similar way. The mole fractions can be calculated from the mass fractions using the molar masses of methanol and water.
03

Calculating Molar Flow Rates

Molar flow rates can be computed by dividing the mass flow rates by the respective molar masses. For methanol and water in the bottom product stream, divide their mass flow rates calculated in the previous step by their respective molar masses.
04

Hypothesizing Sources of Discrepancy

Suppose the mole fraction of methanol in the analyzed bottom product stream is found to be higher than the value calculated. This could be due to a number of reasons, including: high measurement error, volatile conditions in the distillation column causing irregularities, assumptions of ideal behavior not holding, and imprecision in control of the input flow rate.

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Key Concepts

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

Chemical Engineering Education
Understanding the practical applications of chemical processes is a crucial element of chemical engineering education. One of the fundamental processes that chemical engineers need to grasp is distillation, a separation technique that leverages differences in component volatilities. Chemical engineering curriculums emphasize learning through problem-solving and real-world examples, such as the distillation column exercise.

By dissecting the components of the distillation process in such examples鈥攊nput mixtures, distillation conditions, product streams, and their compositions鈥攕tudents develop a thorough understanding of the mass balance concepts that govern chemical engineering operations. The ability to draw process flowcharts and conduct degree-of-freedom analyses equips students with the analytical skills needed to tackle complex engineering problems.
Mass and Mole Fractions
Mass and mole fractions are vital concepts in chemical engineering, representing the ratio of a component's mass or moles to the total mass or moles of the mixture. To illustrate, in the given exercise, the mass fraction of methanol is found by dividing the mass of methanol by the total mass of the bottom stream.

However, for many calculations and analyses, mole fractions are more informative since they facilitate stoichiometric and thermodynamic assessments. To convert mass fractions to mole fractions, one needs to use the molar mass of the components: \[ \text{Mole fraction of component} = \frac{(\text{Mass fraction of component}) \times \text{Total mass}}{\text{Molar mass of component}} \]
Understanding how to perform these conversions is essential for chemical engineers to characterize mixtures and design separation processes accurately.
Process Flowcharting
A flowchart is a graphical representation of a process, displaying the various steps and the flow of materials or information through these steps. In the context of chemical engineering, a process flowchart serves as a vital tool for visualizing and analyzing the steps involved in industrial processes. For instance, the exercise requires drawing a flowchart for a distillation process. This includes
  • An input stream
  • Streams leaving the distillation column (top and bottom)
  • Stream compositions and flow rates

This visual representation aids in identifying the components of the system and is foundational for performing a thorough degree-of-freedom analysis.
Degree-of-Freedom Analysis
Performing a degree-of-freedom analysis is an important step in problem-solving for chemical engineers. This analysis helps to determine whether enough information is available to solve a set of process equations. To perform the analysis, one must count the variables (unknowns) and available equations (relationships between the variables) in the system.

In the provided exercise, this involves considering mass balances of each component and the total mass balance. If the number of equations equals the number of unknowns, the system is solvable. A mismatch suggests that additional information is needed or there are redundancies to address. This concept reinforces the rigorous analytical mindset that is critical for successful process engineering.

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Most popular questions from this chapter

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.

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).

A liquid mixture containing 30.0 mole \(\%\) benzene \((\mathrm{B}), 25.0 \%\) toluene \((\mathrm{T}),\) and the balance xylene \((\mathrm{X})\) is fed to a distillation column. The bottoms product contains 98.0 mole \(\% \mathrm{X}\) and no \(\mathrm{B},\) and \(96.0 \%\) of the \(\mathrm{X}\) in the feed is recovered in this stream. The overhead product is fed to a second column. The overhead product from the second column contains \(97.0 \%\) of the \(\mathrm{B}\) in the feed to this column. The composition of this stream is 94.0 mole\% B and the balance T. (a) Draw and label a flowchart of this process and do the degree-of-freedom analysis to prove that for an assumed basis of calculation, molar flow rates and compositions of all process streams can be calculated from the given information. Write in order the equations you would solve to calculate unknown process variables. In each equation (or pair of simultaneous equations), circle the variable(s) for which you would solve. Do not do the calculations. (b) Calculate (i) the percentage of the benzene in the process feed (i.e., the feed to the first column) that emerges in the overhead product from the second column and (ii) the percentage of toluene in the process feed that emerges in the bottom product from the second column.

A Claus plant converts gaseous sulfur compounds to elemental sulfur, thereby eliminating emission of sulfur into the atmosphere. The process can be especially important in the gasification of coal, which contains significant amounts of sulfur that is converted to \(\mathrm{H}_{2}\) S during gasification. In the Claus process, the \(\mathrm{H}_{2}\) S-rich product gas recovered from an acid-gas removal system following the gasifier is split, with one-third going to a furnace where the hydrogen sulfide is burned at 1 atm with a stoichiometric amount of air to form SO \(_{2}\). $$\mathrm{H}_{2} \mathrm{S}+\frac{3}{2} \mathrm{O}_{2} \rightarrow \mathrm{SO}_{2}+\mathrm{H}_{2} \mathrm{O}$$ The hot gases leave the furnace and are cooled prior to being mixed with the remainder of the \(\mathrm{H}_{2}\) S-rich gases. The mixed gas is then fed to a catalytic reactor where hydrogen sulfide and \(\mathrm{SO}_{2}\) react to form elemental sulfur. $$2 \mathrm{H}_{2} \mathrm{S}+\mathrm{SO}_{2} \rightarrow 2 \mathrm{H}_{2} \mathrm{O}+3 \mathrm{S}$$ The coal available to the gasification process is 0.6 wt\% sulfur, and you may assume that all of the sulfur is converted to \(\mathrm{H}_{2} \mathrm{S}\), which is then fed to the Claus plant. (a) Estimate the feed rate of air to the Claus plant in \(\mathrm{kg} / \mathrm{kg}\) coal. (b) While the removal of sulfur emissions to the atmosphere is environmentally beneficial, identify an environmental concern that still must be addressed with the products from the Claus plant.

Natural gas containing a mixture of methane, ethane, propane, and butane is burned in a furnace with excess air. (a) One hundred kmol/h of a gas containing 94.4 mole\% methane, 3.40\% ethane, 0.60\% propane, and \(0.50 \%\) butane is to be burned with \(17 \%\) excess air. Calculate the required molar flow rate of the air. (b) Let Derive an expression for \(\dot{n}_{\mathrm{a}}\) in terms of the other variables. Check your formula with the results of Part (a). (c) Suppose the feed rate and composition of the fuel gas are subject to periodic variations, and a process control computer is to be used to adjust the flow rate of air to maintain a constant percentage excess. A calibrated electronic flowmeter in the fuel gas line transmits a signal \(R_{\mathrm{f}}\) that is directly proportional to the flow rate \(\left(\dot{n}_{\mathrm{f}}=\alpha R_{\mathrm{f}}\right),\) with a flow rate of \(75.0 \mathrm{kmol} / \mathrm{h}\) yielding a signal \(R_{f}=60 .\) The fuel gas composition is obtained with an on-line gas chromatograph. A sample of the gas is injected into the gas chromatograph (GC), and signals \(A_{1}, A_{2}, A_{3},\) and \(A_{4},\) which are directly proportional to the moles of methane, ethane, propane, and butane, respectively, in the sample, are transmitted. (Assume the same proportionality constant for all species.) The control computer processes these data to determine the required air flow rate and then sends a signal \(R_{\mathrm{a}}\) to a control valve in the air line. The relationship between \(R_{\mathrm{a}}\) and the resulting air flow rate, \(\dot{n}_{\mathrm{a}},\) is another direct proportionality, with a signal \(R_{\mathrm{a}}=25\) leading to an air flow rate of \(550 \mathrm{kmol} / \mathrm{h}\). Write a spreadsheet or computer program to perform the following tasks: (i) Take as input the desired percentage excess and values of \(R_{\mathrm{f}}, A_{1}, A_{2}, A_{3},\) and \(A_{4}\) (ii) Calculate and print out \(\dot{n}_{\mathrm{f}}, x_{1}, x_{2}, x_{3}, x_{4}, \dot{n}_{\mathrm{a}},\) and \(R_{\mathrm{a}}\) Test your program on the data given below, assuming that \(15 \%\) excess air is required in all cases. Then explore the effects of variations in \(P_{\mathrm{xs}}\) and \(R_{\mathrm{f}}\) on \(\dot{n}_{\mathrm{a}}\) for the values of \(A_{1}-A_{4}\) given on the third line of the data table. Briefly explain your results. (d) Finally, suppose that when the system is operating as described, stack gas analysis indicates that the air feed rate is consistently too high to achieve the specified percentage excess. Give several possible explanations.
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