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Chapter 4: Problem 80

Methanol is synthesized from carbon monoxide and hydrogen in a catalytic reactor. The fresh feed to the process contains 32.0 mole \(\%\) CO, \(64.0 \%\) H \(_{2}\), and \(4.0 \%\) Ne. This stream is mixed with a recycle stream in a ratio 5 mol recycle/ 1 mol fresh feed to produce the feed to the reactor, which contains 13.0 mole\% \(\mathrm{N}_{2}\). A low single-pass conversion is attained in the reactor. The reactor effluent goes to a condenser from which two streams emerge: a liquid product stream containing essentially all the methanol formed in the reactor, and a gas stream containing all the \(\mathrm{CO}, \mathrm{H}_{2}\), and \(\mathrm{N}_{2}\) leaving the reactor. The gas stream is split into two fractions: one is removed from the process as a purge stream, and the other is the recycle stream that combines with the fresh feed to the reactor. (a) Assume a methanol production rate of \(100 \mathrm{kmol} / \mathrm{h}\). Perform the DOF for the overall system and all subsystems to prove that there is insufficient information to solve for all unknowns. (b) Briefly explain in your own words the reasons for including (i) the recycle stream and (ii) the purge stream in the process design.

Short Answer

Expert verified
The Degrees of Freedom for the overall system is 1, meaning there is not enough information to solve for all unknowns in this system. The recycling stream is included in the process design to increase the conversion of reactants and to allow unreacted feed from the reactor effluent to be reused. Whereas, the purge stream helps in preventing the build-up of inert materials in the process.

Step by step solution

01

Degrees of Freedom Analysis

To determine the degrees of freedom, you need to know the number of unknown variables ('n') and the number of independent equations ('m'). The degrees of freedom (DOF) is calculated as n - m. For the overall system, there are 10 unknowns: fresh feed rate, recycle feed rate, total reactor feed rate, total effluent feed rate, total condenser outlet feed rate, total purge feed rate, \( \mathrm{CO}, \mathrm{H}_{2}, \mathrm{Ne}, \mathrm{N}_{2} \) in the reactor feed and \( \mathrm{CO_, H}_{2}, \mathrm{Ne}, \mathrm{N}_{2} \) in the recycle stream. We have 9 independent equations: 4 from the component balances around the reactor (for \( \mathrm{CO}, \mathrm{H}_{2}, \mathrm{Ne}, \mathrm{N}_{2} \)), 4 from component balances around the overall system (for \( \mathrm{CO}, \mathrm{H}_{2}, \mathrm{Ne}, \mathrm{N}_{2} \)) and 1 equation from the ratio of the recycle to fresh feed rate. Thus, overall system has DOF = 10 - 9 = 1. This means there isn't sufficient information to solve for all unknowns.
02

Understanding the Role of Recycle and Purge Streams

(i) The recycle stream is used in reactors as it not only increases the conversion of the reactants but also allows unreacted feed from the reactor effluent to be reused in the reactor, providing a sustainable and economical process. (ii) The purge stream is necessary because without it, the inert components (Ne and N2) would accumulate in the recycle stream, decreasing the overall efficiency of the reactor. The purge stream allows us to control the concentration of these inerts in the reactor feed.

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Key Concepts

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

Methanol Synthesis
Methanol synthesis is a critical chemical process where methanol (CH extsubscript{3}OH) is produced using carbon monoxide (CO) and hydrogen (H extsubscript{2}). This process is typically carried out in a catalytic reactor. In our example, the reactor achieves a low single-pass conversion rate. This means that only a small portion of the CO and H extsubscript{2} is converted into methanol during each pass through the reactor.

By-products and unreacted materials are also part of the process, which necessitates handling further downstream or in process optimization strategies. The primary objective is improving efficiency while minimizing waste and operational costs. The methanol synthesis process is vital in producing methanol, which is used in various applications, including fuel, solvents, and antifreeze.
Recycle Stream
In the methanol synthesis process, a recycle stream plays an essential role. This stream recirculates the unreacted gases back into the reactor, increasing the chances they will be converted into methanol.
  • Enhances the overall conversion rate by making the process more economical and efficient.
  • Reduces the consumption of fresh feed, thus lowering operational costs.
Increasing the amount of material going through the reactor without adding fresh feed is beneficial for the reactor鈥檚 performance and product yield. This process step demonstrates a method of efficiently utilizing resources, making it a cornerstone of sustainable chemical engineering practices.
Purge Stream
The purge stream is a crucial component in maintaining process efficiency. Inert gases like Neon (Ne) and Nitrogen (N extsubscript{2}) cannot be converted into methanol, so their accumulation can hinder reactor efficiency.

A purge stream ensures that a portion of the gaseous output is removed from the cycle. This action helps to:
  • Maintain a balanced composition of reactive and inert gases within the reactor.
  • Prevent the buildup of inerts that can lower reaction efficiency.
  • Optimize the reactor's operational conditions by controlling gas compositions effectively.
By carefully managing the purge stream, chemical engineers can maintain optimal conditions within the reactor which is crucial for process efficiency and cost-effectiveness.
Degrees of Freedom Analysis
Degrees of freedom analysis is a mathematical approach used to determine how many variables in a process can be independently controlled. In our methanol synthesis example, there are 10 unknowns and only 9 independent equations, resulting in 1 degree of freedom.

This scenario indicates insufficient information to uniquely solve for all unknowns. Key aspects include:
  • Fresh feed rate and recycle feed rate are among the variables that aren't fully determined due to insufficient equations.
  • Understanding this analysis helps in identifying the constraints and limits of the process design.
Degrees of freedom analysis is instrumental in process design, highlighting where additional information or assumptions are needed to achieve a complete system model.
Component Balances
Component balances are critical calculations in chemical process design to ensure mass conservation across all system and subsystem boundaries. Each chemical species like CO, H extsubscript{2}, N extsubscript{2}, and Ne must be balanced across each unit operation.

In our methanol synthesis process, component balances help us keep track of each substance throughout the system:
  • Account for all input, output, and recycle streams.
  • Crucial in identifying accumulation of inerts or by-products, which can impede process efficiency.
  • Serve as the foundation for developing accurate mathematical models used in process simulation and design.
Maintaining these balances ensures that the process operates as intended, with predictable and optimized results.

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Most popular questions from this chapter

A \(100 \mathrm{kmol} / \mathrm{h}\) stream that is 97 mole \(\%\) carbon tetrachloride \(\left(\mathrm{CCl}_{4}\right)\) and \(3 \%\) carbon disulfide \(\left(\mathrm{CS}_{2}\right)\) is to be recovered from the bottom of a distillation column. The feed to the column is 16 mole \(\% \mathrm{CS}_{2}\) and \(84 \% \mathrm{CCl}_{4},\) and \(2 \%\) of the \(\mathrm{CCl}_{4}\) entering the column is contained in the overhead stream leaving the top of the column. (a) Draw and label a flowchart of the process and do the degree-of-freedom analysis. (b) Calculate the mass and mole fractions of \(\mathrm{CCl}_{4}\) in the overhead stream, and determine the molar flow rates of \(\mathrm{CCl}_{4}\) and \(\mathrm{CS}_{2}\) in the overhead and feed streams. (c) Suppose the overhead stream is analyzed and the mole fraction of \(\mathrm{CS}_{2}\) is found to be significantly lower than the value calculated in Part (b). List as many reasons as you can for the discrepancy, including possible violations of assumptions made in Part (b).

Methane and oxygen react in the presence of a catalyst to form formaldehyde. In a parallel reaction, methane is oxidized to carbon dioxide and water: $$\begin{aligned} \mathrm{CH}_{4}+\mathrm{O}_{2} & \rightarrow \mathrm{HCHO}+\mathrm{H}_{2} \mathrm{O} \\ \mathrm{CH}_{4}+2 \mathrm{O}_{2} & \rightarrow \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O} \end{aligned}$$ The feed to the reactor contains equimolar amounts of methane and oxygen. Assume a basis of \(100 \mathrm{mol}\) feed/s. (a) Draw and label a flowchart. Use a degree-of-freedom analysis based on extents of reaction to determine how many process variable values must be specified for the remaining variable values to be calculated. (b) Use Equation 4.6-7 to derive expressions for the product stream component flow rates in terms of the two extents of reaction, \(\xi_{1}\) and \(\xi_{2}\) (c) The fractional conversion of methane is 0.900 and the fractional yield of formaldehyde is 0.855 . Calculate the molar composition of the reactor output stream and the selectivity of formaldehyde production relative to carbon dioxide production. (d) A classmate of yours makes the following observation: "If you add the stoichiometric equations for the two reactions, you get the balanced equation $$2 \mathrm{CH}_{4}+3 \mathrm{O}_{2} \rightarrow \mathrm{HCHO}+\mathrm{CO}_{2}+3 \mathrm{H}_{2} \mathrm{O}$$ The reactor output must therefore contain one mole of \(\mathrm{CO}_{2}\) for every mole of HCHO, so the selectivity of formaldehyde to carbon dioxide must be \(1.0 .\) Doing it the way the book said to do it, \(I\) got a different selectivity. Which way is right, and why is the other way wrong?" What is your response?

Ammonia is oxidized to nitric oxide in the following reaction: $$4 \mathrm{NH}_{3}+5 \mathrm{O}_{2} \rightarrow 4 \mathrm{NO}+6 \mathrm{H}_{2} \mathrm{O}$$ (a) Calculate the ratio (lb-mole \(\mathrm{O}_{2}\) react/lb-mole NO formed). (b) If ammonia is fed to a continuous reactor at a rate of \(100.0 \mathrm{kmol} \mathrm{NH}_{3} / \mathrm{h}\), what oxygen feed rate (kmol/h) would correspond to 40.0\% excess O_? (c) If \(50.0 \mathrm{kg}\) of ammonia and \(100.0 \mathrm{kg}\) of oxygen are fed to a batch reactor, determine the limiting reactant, the percentage by which the other reactant is in excess, and the extent of reaction and mass of NO produced (kg) if the reaction proceeds to completion.

In an absorption tower (or absorber), a gas is contacted with a liquid under conditions such that one or more species in the gas dissolve in the liquid. A stripping tower (or stripper) also involves a gas contacting a liquid, but under conditions such that one or more components of the feed liquid come out of solution and exit in the gas leaving the tower. A process consisting of an absorption tower and a stripping tower is used to separate the components of a gas containing 30.0 mole \(\%\) carbon dioxide and the balance methane. A stream of this gas is fed to the bottom of the absorber. A liquid containing 0.500 mole\% dissolved \(\mathrm{CO}_{2}\) and the balance methanol is recycled from the bottom of the stripper and fed to the top of the absorber. The product gas leaving the top of the absorber contains 1.00 mole \(\% \mathrm{CO}_{2}\) and essentially all of the methane fed to the unit. The CO_-rich liquid solvent leaving the bottom of the absorber is fed to the top of the stripper and a stream of nitrogen gas is fed to the bottom. Ninety percent of the \(\mathrm{CO}_{2}\) in the liquid feed to the stripper comes out of solution in the column, and the nitrogen/CO_stream leaving the column passes out to the atmosphere through a stack. The liquid stream leaving the stripping tower is the \(0.500 \% \mathrm{CO}_{2}\) solution recycled to the absorber. The absorber operates at temperature \(T_{\mathrm{a}}\) and pressure \(P_{\mathrm{a}}\) and the stripper operates at \(T_{\mathrm{s}}\) and \(P_{\mathrm{s}}\) Methanol may be assumed to be nonvolatile- -that is, none enters the vapor phase in either column and \(\mathrm{N}_{2}\), may be assumed insoluble in methanol. (a) In your own words, explain the overall objective of this two-unit process and the functions of the absorber and stripper in the process. (b) The streams fed to the tops of each tower have something in common, as do the streams fed to the bottoms of each tower. What are these commonalities and what is the probable reason for them? (c) Taking a basis of 100 mol/h of gas fed to the absorber, draw and label a flowchart of the process. For the stripper outlet gas, label the component molar flow rates rather than the total flow rate and mole fractions. Do the degree-of-freedom analysis and write in order the equations you would solve to determine all unknown stream variables except the nitrogen flow rate entering and leaving the stripper. Circle the variable(s) for which you would solve each equation (or set of simultaneous equations), but don't do any of the calculations yet. (d) Calculate the fractional \(\mathrm{CO}_{2}\) removal in the absorber (moles absorbed/mole in gas feed) and the molar flow rate and composition of the liquid feed to the stripping tower. (e) Calculate the molar feed rate of gas to the absorber required to produce an absorber product gas flow rate of \(1000 \mathrm{kg} / \mathrm{h}\). (f) Would you guess that \(T_{\mathrm{s}}\) would be higher or lower than \(T_{\mathrm{a}} ?\) Explain. (Hint: Think about what happens when you heat a carbonated soft drink and what you want to happen in the stripper.) What about the relationship of \(P_{\mathrm{s}}\) to \(P_{\mathrm{a}} ?\) (g) What properties of methanol would you guess make it the solvent of choice for this process? (In more general terms, what would you look for when choosing a solvent for an absorption-stripping process to separate one gas from another?)

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.
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