/*! 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

In an attempt to conserve water and to be awarded LEED (Leadership in Energy and Environmental Design) certification, a 20,000-liter cistem has been installed during construction of a new building. The cistem collects water from an HVAC (heating, ventilation, and air-conditioning) system designed to provide 2830 cubic meters of air per minute at \(22^{\circ} \mathrm{C}\) and \(50 \%\) relative humidity after converting it from ambient conditions \(\left(31^{\circ} \mathrm{C}, 70 \% \text { relative humidity }\right) .\) The collected condensate serves as the source of water for lawn maintenance. Estimate (a) the rate of intake of air at ambient conditions in cubic feet per minute and (b) the hours of operation required to fill the cistern.

Nitric acid is a chemical intermediate primarily used in the synthesis of ammonium nitrate, which is used in the manufacture of fertilizers. The acid also is important in the production of other nitrates and in the separation of metals from ores. Nitric acid may be produced by oxidizing ammonia to nitric oxide over a platinum-rhodium catalyst, then oxidizing the nitric oxide to nitrogen dioxide in a separate unit where it is absorbed in water to form an aqueous solution of nitric acid.The reaction sequence is as follows:$$\begin{aligned} 4 \mathrm{NH}_{3}+5 \mathrm{O}_{2} & \rightarrow 4 \mathrm{NO}+6 \mathrm{H}_{2} \mathrm{O} \\\4 \mathrm{NO}+2 \mathrm{O}_{2} & \rightarrow 4 \mathrm{NO}_{2} \\\4 \mathrm{NO}_{2}+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})+\mathrm{O}_{2} & \rightarrow 4 \mathrm{HNO}_{3}(\mathrm{aq}) \end{aligned}$$.Ammonia vapor produced by vaporizing pure liquid ammonia at 820 kPa absolute is mixed with air, and the combined stream enters the ammonia oxidation unit. Air at \(30^{\circ} \mathrm{C}, 1\) atm absolute, and \(50 \%\) relative humidity is compressed and fed to the process. A fraction of the air is sent to the cooling and hydration units, while the remainder is passed through a heat exchanger and mixed with the ammonia. The total oxygen fed to the process is the amount stoichiometrically required to convert all of the ammonia to HNO \(_{3},\) while the fraction sent to the ammonia oxidizer corresponds to the stoichiometric amount required to convert ammonia to NO.The ammonia reacts completely in the oxidizer, with \(97 \%\) forming NO and the rest forming \(\mathrm{N}_{2}\). Only a negligible amount of \(\mathrm{NO}_{2}\) is formed in the oxidizer. However, the gas leaving the oxidizer is subjected to a series of cooling and hydration steps in which the NO is completely oxidized to \(\mathrm{NO}_{2}\) which in turn combines with water (some of which is present in the gas from the oxidizer and the rest is added) to form a 55 wt\% aqueous solution of nitric acid. The product gas from the process may be taken to contain only \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\). (a) Taking a basis of \(100 \mathrm{kmol}\) of ammonia fed to the process, calculate (i) the volumes \(\left(\mathrm{m}^{3}\right)\) of the ammonia vapor and air fed to the process using the compressibility-factor equation of state; (ii) the amount (kmol) and composition (in mole fractions) of the gas leaving the oxidation unit; (iii) the required volume of liquid water \(\left(\mathrm{m}^{3}\right)\) that must be fed to the cooling and hydration units; and (iv) the fraction of the air fed to the ammonia oxidizer. (b) Scale the results from Part (a) to a new basis of 100 metric tons per hour of 55\% nitric acid solution.(c) Nitrogen oxides (collectively referred to as \(\mathrm{NO}_{x}\) ) are a category of pollutants that are formed in many ways, including processes like that described in this problem. List the annual emission rates of the three largest sources of \(\mathrm{NO}_{x}\) emissions in your home region. What are the effects of exposure to excessive concentrations of \(\mathrm{NO}_{x} ?\) (d) A platinum-rhodium catalyst is used in ammonia oxidation. Fxplain the function of the catalyst, describe its structure, and explain the relationship of the structure to the function.

A stage of a separation process is defined as an operation in which components of one or more feed streams divide themselves between two phases, and the phases are taken off separately. In an ideal stage or equilibrium stage, the effluent (exit) streams are in equilibrium with each other.Distillation columns often consist of a series of vertically distributed stages. Vapor flows upward and liquid flows downward between adjacent stages; some of the liquid fed to each stage vaporizes,and some of the vapor fed to each stage condenses. A representation of a section of a distillation column is shown below. (See Problem 4.42 for a more realistic representation.) Consider a distillation column operating at 0.4 atm absolute in which benzene and styrene are being separated. A vapor stream containing 65 mole\% benzene and 35 mole\% styrene enters stage 1 at a rate of \(200 \mathrm{mol} / \mathrm{h}\), and liquid containing 55 mole\% benzene and 45 mole\% styrene leaves this stage at a rate of 150 mol/h. You may assume (1) the stages are ideal, (2) Raoult's law can be used to relate the compositions of the streams leaving each stage, and (3) the total vapor and liquid molar flow rates do not change by a significant amount from one stage to the next.(a) How would you expect the mole fraction of benzene in the liquid to vary from one stage to another, beginning with stage 1 and moving up the column? In light of your answer and considering that the pressure remains essentially constant from one stage to another, how would you then expect the temperature to vary at progressively higher stages? Briefly explain. (b) Estimate the temperature at stage 1 and the compositions of the vapor stream leaving this stage and the liquid stream entering it. Then repeat these calculations for stage 2 . (c) Describe how you would calculate the number of ideal stages required to reduce the styrene content of the vapor to less than 5 mole\%.

A fuel gas containing methane and ethane is burned with air in a furnace, producing a stack gas at \(300^{\circ} \mathrm{C}\) and \(105 \mathrm{kPa}\) (absolute). You analyze the stack gas and find that it contains no unburned hydrocarbons, oxygen, or carbon monoxide. You also determine the dew-point temperature.(a) Estimate the range of possible dew-point temperatures by determining the dew points when the feed is either pure methane or pure ethane. (b) Estimate the fraction of the feed that is methane if the measured dew- point temperature is \(59.5^{\circ} \mathrm{C}\). (c) What range of measured dew point temperatures would lead to calculated methane mole fractions within 5\% of the value determined in Part (b)?

In-Hexane is used to extract oil from soybeans. (See Problem 6.24 .) The solid residue from the extraction unit, which contains 0.78 kg liquid hexane/kg dry solids, is contacted in a dryer with nitrogen that enters at \(85^{\circ} \mathrm{C}\). The solids leave the dryer containing \(0.05 \mathrm{kg}\) liquid hexane/kg dry solids, and the gas leaves the dryer at \(80^{\circ} \mathrm{C}\) and 1.0 atm with a relative saturation of \(70 \% .\) The gas is then fed to a condenser in which it is compressed to 5.0 atm and cooled to \(28^{\circ} \mathrm{C}\), enabling some of the hexane to be recovered as condensate.(a) Calculate the fractional recovery of hexane (kg condensed/kg fed in wet solids). (b) A proposal has been made to split the gas stream leaving the condenser, combining 90\% of it with fresh makeup nitrogen, heating the combined stream to \(85^{\circ} \mathrm{C},\) and recycling the heated stream to the dryer inlet. What fraction of the fresh nitrogen required in the process of Part (a) would be saved by introducing the recycle? What costs would be incurred by introducing the recycle?
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