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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

A liquid mixture contains \(60.0 \mathrm{wt} \%\) ethanol \((\mathrm{E}), 5.0 \mathrm{wt} \%\) of a dissolved solute \((\mathrm{S}),\) and the balance water. A stream of this mixture is fed to a continuous distillation column operating at steady state. Product streams emerge at the top and bottom of the column. The column design calls for the product streams to have equal mass flow rates and for the top stream to contain 90.0 wt\% ethanol and no S. (a) Assume a basis of calculation, draw and fully label a process flowchart, do the degree-of-freedom analysis, and verify that all unknown stream flows and compositions can be calculated. (Don't do any calculations yet.) (b) Calculate (i) the mass fraction of \(S\) in the bottom stream and (ii) the fraction of the ethanol in the feed that leaves in the bottom product stream (i.e., \(\mathrm{kg} \mathrm{E}\) in bottom stream/kg \(\mathrm{E}\) in feed) if the process operates as designed. (c) An analyzer is available to determine the composition of ethanol-water mixtures. The calibration curve for the analyzer is a straight line on a plot on logarithmic axes of mass fraction of ethanol, \(x\) (kg E/kg mixture), versus analyzer reading, \(R\). The line passes through the points \((R=15, x=\) 0.100) and \((R=38, x=0.400)\). Derive an expression for \(x\) as a function of \(R(x=\cdots\) ) based on the calibration, and use it to determine the value of \(R\) that should be obtained if the top product stream from the distillation column is analyzed. (d) Suppose a sample of the top stream is taken and analyzed and the reading obtained is not the one calculated in Part (c). Assume that the calculation in Part (c) is correct and that the plant operator followed the correct procedure in doing the analysis. Give five significantly different possible causes for the deviation between \(R_{\text {measured and }} R_{\text {prediced }}\), including several assumptions made when writing the balances of Part (c). For each one, suggest something that the operator could do to check whether it is in fact the problem.

A liquid-phase chemical reaction \(\mathrm{A} \rightarrow \mathrm{B}\) takes place in a well-stirred tank. The concentration of \(\mathrm{A}\) in the feed is \(C_{\mathrm{A} 0}\left(\operatorname{mol} / \mathrm{m}^{3}\right),\) and that in the tank and outlet stream is \(C_{\mathrm{A}}\left(\mathrm{mol} / \mathrm{m}^{3}\right) .\) Neither concentration varies with time. The volume of the tank contents is \(V\left(\mathrm{m}^{3}\right)\) and the volumetric flow rate of the inlet and outlet streams is \(\dot{V}\left(\mathrm{m}^{3} / \mathrm{s}\right)\). The reaction rate (the rate at which \(\mathrm{A}\) is consumed by reaction in the tank) is given by the expression $$r(\text { mol } A \text { consumed } / \mathrm{s})=k V C_{\mathrm{A}}$$ (a) Is this process continuous, batch, or semibatch? Is it transient or steady-state? (b) What would you expect the reactant concentration \(C_{\mathrm{A}}\) to equal if \(k=0\) (no reaction)? What should it approach if \(k \rightarrow \infty\) (infinitely rapid reaction)? (c) Write a differential balance on \(A,\) stating which terms in the general balance equation (accumulation = input + generation - output - consumption) you discarded and why you discarded them. Use the balance to derive the following relation between the inlet and outlet reactant concentrations: $$C_{\mathrm{A}}=\frac{C_{\mathrm{A} 0}}{1+k V / \dot{V}}$$ Verify that this relation predicts the results in Part (b).

In the production of a bean oil, beans containing 13.0 wt\% oil and \(87.0 \%\) solids are ground and fed to a stirred tank (the extractor) along with a recycled stream of liquid \(n\) -hexane. The feed ratio is \(3 \mathrm{kg}\) hexane/kg beans. The ground beans are suspended in the liquid, and essentially all of the oil in the beans is extracted into the hexane. The extractor effluent passes to a filter where the solids are collected and form a filter cake. The filter cake contains 75.0 wt\% bean solids and the balance bean oil and hexane, the latter two in the same ratio in which they emerge from the extractor. The filter cake is discarded and the liquid filtrate is fed to a heated evaporator in which the hexane is vaporized and the oil remains as a liquid. The oil is stored in drums and shipped. The hexane vapor is subsequently cooled and condensed, and the liquid hexane condensate is recycled to the extractor. (a) Draw and label a flowchart of the process, do the degree-of-freedom analysis, and write in an efficient order the equations you would solve to determine all unknown stream variables, circling the variables for which you would solve. (b) Calculate the yield of bean oil product (kg oil/kg beans fed), the required fresh hexane feed \(\left(\mathrm{kg} \mathrm{C}_{6} \mathrm{H}_{14} / \mathrm{kg} \text { beans fed }\right),\) and the recycle to fresh feed ratio (kg hexane recycled/kg fresh feed). (c) It has been suggested that a heat exchanger might be added to the process. This process unit would consist of a bundle of parallel metal tubes contained in an outer shell. The liquid filtrate would pass from the filter through the inside of the tubes and then go on to the evaporator. The hot hexane vapor on its way from the evaporator to the extractor would flow through the shell, passing over the outside of the tubes and heating the filtrate. How might the inclusion of this unit lead to a reduction in the operating cost of the process? (d) Suggest additional steps that might improve the process economics.

A liquid mixture of acetone and water contains 35 mole\% acetone. The mixture is to be partially evaporated to produce a vapor that is 75 mole \(\%\) acetone and leave a residual liquid that is 18.7 mole \(\%\) (a) Suppose the process is to be carried out continuously and at steady state with a feed rate of 10.0 kmol/h. Let \(\dot{n}_{\mathrm{v}}\) and \(\dot{n}_{1}\) be the flow rates of the vapor and liquid product streams, respectively. Draw and label a process flowchart, then write and solve balances on total moles and on acetone to determine the values of \(\dot{n}_{\mathrm{v}}\) and \(\dot{n}_{\mathrm{l}}\). For each balance, state which terms in the general balance equation (accumulation \(=\)input \(+\)generation \(-\)output\(-\)consumption ) can be discarded and why. (See Example 4.2-2.) (b) Now suppose the process is to be carried out in a closed container that initially contains 10.0 kmol of the liquid mixture. Let \(n_{\mathrm{v}}\) and \(n_{1}\) be the moles of final vapor and liquid phases, respectively. Draw and label a process flowchart, then write and solve integral balances on total moles and on acetone. For each balance, state which terms of the general balance equation can be discarded and why. (c) Returning to the continuous process, suppose the vaporization unit is built and started and the product stream flow rates and compositions are measured. The measured acetone content of the vapor stream is 75 mole \(\%\) acetone, and the product stream flow rates have the values calculated in Part (a). However, the liquid product stream is found to contain 22.3 mole \(\%\) acetone. It is possible that there is an error in the measured composition of the liquid stream, but give at least five other reasons for the discrepancy. [Think about assumptions made in obtaining the solution of Part (a).]

Inside a distillation column (see Problem 4.8), a downward-flowing liquid and an upward-flowing vapor maintain contact with each other. For reasons we will discuss in greater detail in Chapter \(6,\) the vapor stream becomes increasingly rich in the more volatile components of the mixture as it moves up the column, and the liquid stream is enriched in the less volatile components as it moves down. The vapor leaving the top of the column goes to a condenser. A portion of the condensate is taken off as a product (the overhead product), and the remainder (the reflux) is returned to the top of the column to begin its downward journey as the liquid stream. The condensation process can be represented as shown below: A distillation column is being used to separate a liquid mixture of ethanol (more volatile) and water (less volatile). A vapor mixture containing 89.0 mole \(\%\) ethanol and the balance water enters the overhead condenser at a rate of \(100 \mathrm{lb}\) -mole/h. The liquid condensate has a density of \(49.01 \mathrm{b}_{\mathrm{m}} / \mathrm{ft}^{3},\) and the reflux ratio is \(3 \mathrm{lb}_{\mathrm{m}}\) reflux/lb \(_{\mathrm{m}}\) overhead product. When the system is operating at steady state, the tank collecting the condensate is half full of liquid and the mean residence time in the tank (volume of liquid/volumetric flow rate of liquid) is 10.0 minutes. Determine the overhead product volumetric flow rate (ft \(^{3}\) /min) and the condenser tank volume (gal).
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