/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 A vapor mixture of \(n\) -butane... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 62

A vapor mixture of \(n\) -butane (B) and \(n\) -hexane (H) contains 50.0 mole\% butane at \(120^{\circ} \mathrm{C}\) and 1.0 atm. A stream of this mixture flowing at a rate of \(150.0 \mathrm{L} / \mathrm{s}\) is cooled and compressed, causing some but not all of the vapor to condense. (Treat this process as a single-unit operation.) Liquid and vapor product streams emerge from the process in equilibrium at \(T\left(^{\circ} \mathrm{C}\right)\) and \(1100 \mathrm{mm} \mathrm{Hg}\). The vapor product contains 60.0 mole\% butane.(a) Draw and label a flowchart. Perform a degree-of-freedom analysis to show that you have enough information to determine the required final temperature ( \(T\) ), the composition of the liquid product (component mole fractions), and the molar flow rates of the liquid and vapor products from the given information and Antoine expressions for the vapor pressures \(p_{\mathrm{B}}^{*}(T)\) and \(p_{\mathrm{H}}^{*}(T) .\) Just identify the equations - for example, mole balance on butane or Raoult's law for hexane-but don't write them yet.(b) Write in order the equations you would use to determine the quantities listed in Part (a) and also the fractional condensation of hexane (mol \(\mathrm{H}\) condensed/mol \(\mathrm{H}\) fed). In each equation, circle the variable for which you would solve. Do no algebra or calculations.(c) Complete the calculations either manually or with an equation-solving program.(d) State three assumptions you made that could lead to errors in the calculated quantities.

Short Answer

Expert verified
This problem involves a complex combination of concepts in chemical engineering thermodynamics. After setting up and solving the system of equations from principles of material balance and phase equilibrium, assumptions are identified and discussed. Errors could originate from assuming ideal behavior and constant conditions, as well as the accuracy of Antoine's equations at given conditions.

Step by step solution

01

Flowchart and Degree-of-Freedom analysis

Drawing the flowchart can help visualize the problem and organize given and required data. Indicate the known initial compositions, molar flow rates and condition, and mark the points where unknowns (the final temperature T, molar flows and composition of product streams) should be. From the drawn flowchart, a degree-of-freedom analysis can be done by comparing the number of unknowns and available relations, including mole balances around the process for both components, and phase equilibrium relations implied by Raoult's Law applied to butane and hexane.
02

Writing the Required Equations

From the degree-of-freedom analysis, the system was determined as solvable, meaning there exists as many equations as unknowns. The actual equations formulating the problem represent the mole balances which account the butane and hexane entering and exiting the unit, and Raoult's law for butane and hexane in the liquid and vapor outlets. Write each of the equations, with the variable of interest identified and circled in each.
03

Solve the Equations

After properly writing all the equations, proceed to solve the equations for the circled unknowns. This step may require the assistance of an equation-solving software or solve it manually if possible. After solving, go back to each equation to verify the answers.
04

Determine Assumptions and Possible Errors

Given that the problem requires assumptions, identify what assumptions have been made, such as the mixture behaving ideally, negligible changes in molar volumes upon mixing, constant temperature and pressure in the unit operation, and Antoine's equations adequately representing the vapor pressures of the substances at the given conditions. Discuss how these assumptions could lead to errors in the calculations.

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Key Concepts

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

Phase Equilibrium
Phase equilibrium occurs when a substance coexists in different states, like liquid and vapor, without changing over time. In a chemical process, when two phases are in equilibrium, the exchange between them happens at the same rate for a given component. This means no net mass change between phases. In our exercise, phase equilibrium applies as the butane/hexane mixture partially condenses, reaching equilibrium at a certain temperature (T) and pressure (1100 mmHg). This ensures the vapor and liquid streams have a stable composition over time, with fixed mole fractions for each component.

Understanding phase equilibrium in chemical processes is crucial because it impacts separation strategies, energy needs, and product purity. Processes like distillation rely heavily on phase equilibrium for efficient separation of components. Here, identifying the final equilibrium T helps determine how much vapor condenses and the composition of each stream, making it a vital calculation part in process analysis.
Raoult's Law
Raoult's Law provides a way to relate the vapor pressure of a component in a mixture to its mole fraction in the liquid phase. It states that the partial vapor pressure of a component is the product of its mole fraction in the liquid phase and its pure component vapor pressure. This relationship is crucial for describing phase behavior in ideal mixtures.
  • The law applies when considering how a substance behaves in a mixture, like our butane and hexane vapor.
  • Given our scenario, Raoult's Law allows us to determine how much each component contributes to the total vapor phase using their individual vapor pressures.
  • Knowing the relative amounts of butane and hexane in each phase helps in calculating the distribution between liquid and vapor.
By applying Raoult's Law, we can theoretically predict the vapor composition of butane and hexane at different temperatures and pressures. In our exercise, it directly assists in determining the mole fractions of both components in equilibrium, making it central to achieving accurate calculations of the process outputs.
Degree-of-Freedom Analysis
Degree-of-freedom analysis in chemical processes is a systematic method to determine whether enough information is available to solve a system. It counts variables and equations, balancing them to decide solvability. This concept applies heavily in complex systems, ensuring all unknowns can be uncovered using available data.

For the exercise problem, a degree-of-freedom analysis helps us evaluate if we can find the final temperature, the composition of the liquid product, and molar flow rates. This involves:
  • Listing known variables such as initial compositions, flow rates, and pressures.
  • Determining the number of equations available, like mole balances or phase relations via Raoult's Law.
  • Checking if equations equal the number of unknowns. If they do, as in this problem, solving the system is feasible.
Understanding degree-of-freedom analysis allows engineers to manage complex cases systematically and is essential in ensuring process designs meet specified requirements. It helps pinpoint the parameters needed for successful process completion, like in this partial condensation step.

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Most popular questions from this chapter

The pressure in a vessel containing methane and water at \(70^{\circ} \mathrm{C}\) is 10 atm. At the given temperature, the Henry's law constant for methane is \(6.66 \times 10^{4}\) atm/mole fraction. Estimate the mole fraction of methane in the liquid.

Benzene and hexane are being considered as solvents to extract acetic acid from aqueous mixtures. At \(30^{\circ} \mathrm{C},\) distribution coefficients for the two solvents are \(\mathrm{K}_{\mathrm{B}}=0.098\) mass fraction acetic acid in benzene/mass fraction acetic acid in water and \(\mathrm{K}_{\mathrm{H}}=0.017\) mass fraction acetic acid in hexane/mass fraction acetic acid in water.(a) Based on the distribution coefficients only, which of the two solvents would you use and why? Demonstrate the logic of your decision by comparing the quantities of the two solvents required to reduce the acetic acid content in \(100 \mathrm{kg}\) of an aqueous solution from \(30 \mathrm{wt} \%\) to \(10 \mathrm{wt} \%\).(b) What other factors may be important in choosing between benzene and cyclohexane?

An air conditioner is designed to bring \(10,000 \mathrm{ft}^{3} / \mathrm{min}\) of outside air \(\left(90^{\circ} \mathrm{F}, 29.8 \text { inches } \mathrm{Hg} .88 \%\right.\) relative humidity) to \(40^{\circ} \mathrm{F}\), thereby condensing a portion of the water vapor, and then to reheat the air before releasing it into a room at \(65^{\circ} \mathrm{F}\). Calculate the rate of condensation (gallons \(\mathrm{H}_{2} \mathrm{O} / \mathrm{min}\) ) and the volumetric flow rate of the air delivered to the room. (Suggestion: On the flowchart, treat the coolingcondensation and the reheating as separate process steps.)

A liquid mixture contains \(N\) components ( \(N\) may be any number from 2 to 10 ) at pressure \(P(\mathrm{mm} \mathrm{Hg})\). The mole fraction of the ith component is \(x_{i}(i=1,2, \ldots, N),\) and the vapor pressure of that component is given by the Antoine equation (see Table B.4) with constants \(A_{i}, B_{i},\) and \(C_{i}\). Raoult's law may be applied to each component.(a) Write the equations you would use to calculate the bubble-point temperature of the mixture, ending with an equation of the form \(f(T)=0 .\) (The value of \(T\) that satisfies this equation is the bubble-point temperature.) Then write the equations for the component mole fractions \(\left(y_{1}, y_{2}, \ldots, y_{N}\right)\) in the first bubble that forms, assuming that the temperature is now known.(b) Prepare a spreadsheet to perform the calculations of Part (a). The spreadsheet should include a title line for identification of the problem and a row that has entries for the given pressure and an estimate of the system temperature. Be sure to label these variables and show the units in which each is expressed. Adjacent columns should be headed Species, \(p_{i}^{*}, x_{i}, p_{i},\) and \(y_{i} .\) Values of vapor pressures at the estimated temperature should be calculated using the physical property database in APEx, and Raoult's law should be used to determine partial pressures. The final row in the table should have the sums of the vapor mole fractions and partial pressures. Place a convergence function \(f(T)=P-\Sigma p_{i}\) below the table so that Goal Seek can be used to vary the estimated \(T\) until \(f(T)=0 .\) Test the spreadsheet by calculating the bubble-point temperature for a liquid mixture containing 22.6 mole \(\%\) benzene, \(22.6 \%\) ethylbenzene, \(22.3 \%\) toluene, and the balance styrene at pressures of \(250 \mathrm{mm} \mathrm{Hg}, 760 \mathrm{mm} \mathrm{Hg},\) and \(7500 \mathrm{mm}\) Hg. Identify any concerns you may have about the calculated results.(c) It is determined that instead of styrene, the balance of the above mixture in Part (b) is propylbenzene. Upon entering the name "propylbenzene鈥� in the APEx AntoineP estimator, you probably get the error message #VALUE!, which means that this substance is not in the APEx database. Poling et al. (see Footnote 2, p. A.57) provide constants for the vapor pressure of propylbenzene corresponding to the following expression of the Antoine equation:$$\log _{10} p^{*}(\mathrm{bar})=A-B /\left[T\left(^{\circ} \mathrm{C}\right)+C\right]$$ where \(A=4.07664, B=1491.8,\) and \(C=207.25 ;\) the correlation is valid over the range \(324 \mathrm{K}-\) 461 K. Modify the spreadsheet to incorporate this expression, and estimate the bubble-point temperature of the mixture at a pressure of \(760 \mathrm{mm} \mathrm{Hg}\).

Penicillin is produced by fermentation and recovered from the resulting aqueous broth by extraction with butyl acetate. The penicillin distribution coefficient \(K\) (mass fraction of penicillin in the butyl acetate phase/mass fraction of penicillin in the water phase) depends strongly on the pH in the aqueous phase:$$\begin{array}{|r|c|c|c|}\hline \mathrm{pH} & 2.1 & 4.4 & 5.8 \\\\\hline K & 25.0 & 1.38 & 0.10 \\\\\hline\end{array}$$,This dependence provides the basis for the process to be described. Water and butyl acetate may be considered immiscible. The extraction is performed in the following three-unit process:\(\bullet\) After filtration, broth from a fermentor containing dissolved penicillin, other soluble impurities, and water is acidified in a mixing tank. The acidified broth, which contains 1.5 wt\% penicillin, is contacted with liquid butyl acetate in an extraction unit consisting of a mixer, in which the aqueous and organic phases are brought into intimate contact with each other, followed by a settling tank, in which the two phases separate under the influence of gravity. The pH of the aqueous phase in the extraction unit equals \(2.1 .\) In the mixer \(90 \%\) of the penicillin in the feed broth transfers from the aqueous phase to the organic phase.\(\bullet\) The two streams leaving the settler are in equilibrium with each other- -that is, the ratio of the penicillin mass fractions in the two phases equals the value of \(K\) corresponding to the pH of the aqueous phase \((=2.1 \text { in Unit } 1\) ). The impurities in the feed broth remain in the aqueous phase. The raffinate (by definition, the product stream containing the feed-solution solvent) leaving Extraction Unit 1 is sent elsewhere for further processing, and the organic extract (the product stream containing the extracting solvent) is sent to a second mixer-settler unit.\(\bullet\) In the second unit, the organic solution fed to the mixing stage is contacted with an alkaline aqueous solution that adjusts the pH of the aqueous phase in the unit to \(5.8 .\) In the mixer, \(90 \%\) of the penicillin entering in the organic feed solution transfers to the aqueous phase. Once again, the two streams emerging from the settler are in equilibrium. The aqueous extract is the process product.(a) Taking a basis of \(100 \mathrm{kg}\) of acidified broth fed to the first extraction unit, draw and completely label a flowchart of this process and carry out the degree-of-freedom analysis to show that all labeled variables can be determined. (Suggestion: Consider the combination of water, impurities, and acid as a single species and the alkaline solution as a second single species, since the components of these "pseudospecies" always stay together in the process.)(b) Calculate the ratios (kg butyl acetate required/kg acidified broth) and (kg alkaline solution required/kg acidified broth) and the mass fraction of penicillin in the product solution.(c) Briefly explain the following:(i) What is the likely reason for transferring most of the penicillin from an aqueous phase to an organic phase and then transferring most of it back to an aqueous phase, when each transfer leads to a loss of some of the drug? (ii) What is the purpose of acidifying the broth prior to the first extraction stage, and why is the extracting solution added to the second unit a base? (iii) Why are the two "raffinates" in the process the aqueous phase leaving the first unit and the organic phase leaving the second unit, and vice versa for the "extracts"? (Look again at the definitions of these terms.)(d) An alternative process for recovering the penicillin from the fermentation broth might involve evaporation to dryness. In that case, all the water simply is evaporated. Give two possible reasons for rejection of this alternative.
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