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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

Strawberries contain about \(15 \mathrm{wt} \%\) solids and \(85 \mathrm{wt} \%\) water. To make strawberry jam, crushed strawberries and sugar are mixed in a 45: 55 mass ratio, and the mixture is heated to evaporate water until the residue contains one-third water by mass. (a) Draw and label a flowchart of this process. (b) Do the degree-of-freedom analysis and show that the system has zero degrees of freedom (i.e., the number of unknown process variables equals the number of equations relating them). If you have too many unknowns, think about what you might have forgotten to do. (c) Calculate how many pounds of strawberries are needed to make a pound of jam. (d) Making a pound of jam is something you could accomplish in your own kitchen (or maybe even a dorm room). However, a typical manufacturing line for jam might produce 1500 1b_m/h. List technical and economic factors you would have to take into account as you scaled up this process from your kitchen to a commercial operation.

Ethyl acetate (A) undergoes a reaction with sodium hydroxide (B) to form sodium acetate and ethyl alcohol: The reaction is carried out at steady state in a series of stirred-tank reactors. The output from the ith reactor is the input to the \((i+1)\) st reactor. The volumetric flow rate between the reactors is constant at \(\dot{V}(\mathrm{L} / \mathrm{min}),\) and the volume of each tank is \(V(\mathrm{L})\) The concentrations of \(\mathrm{A}\) and \(\mathrm{B}\) in the feed to the first tank are \(C_{\mathrm{A} 0}\) and \(C_{\mathrm{B} 0}(\mathrm{mol} / \mathrm{L})\). The tanks are stirred sufficiently for their contents to be uniform throughout, so that \(C_{\mathrm{A}}\) and \(C_{\mathrm{B}}\) in a tank equal \(C_{\mathrm{A}}\) and \(C_{\mathrm{B}}\) in the stream leaving that tank. The rate of reaction is given by the expression where \(k[\mathrm{L} /(\mathrm{mol} \cdot \mathrm{min})]\) is the reaction rate constant. (a) Write a material balance on \(A\) in the \(i\) th tank, and show that it yields where \(\tau=V / \dot{V}\) is the mean residence time in each tank. Then write a balance on \(\mathrm{B}\) in the ith tank and subtract the two balances, using the result to prove that (b) Use the equations derived in Part (a) to prove that and from this relation derive an equation of the form where \(\alpha, \beta,\) and \(\gamma\) are functions of \(k, C_{\mathrm{A} 0}, C_{\mathrm{B} 0}, C_{\mathrm{A}, i-1},\) and \(\tau .\) Then write the solution of this equation for \(C_{\mathrm{A} i}\) (c) Write a spreadsheet or computer program to calculate \(N\), the number of tanks needed to achieve a fractional conversion \(x_{\mathrm{A} N} \geq x_{\mathrm{Af}}\) at the outlet of the final reactor. Your program should implement the following procedure: (i) Take as input values of \(k, \dot{V}, V, C_{\mathrm{A} 0}(\mathrm{mol} / \mathrm{L}), C_{\mathrm{B} 0}(\mathrm{mol} / \mathrm{L}),\) and \(x_{\mathrm{Af}}\) (ii) Use the equation for \(C_{A i}\) derived in Part ( \(b\) ) to calculate \(C_{\mathrm{Al}}\); then calculate the corresponding fractional conversion \(x_{\mathrm{A} 1}\) (iii) Repeat the procedure to calculate \(C_{\mathrm{A} 2}\) and \(x_{\mathrm{A} 2},\) then \(C_{\mathrm{A} 3}\) and \(x_{\mathrm{A} 3},\) continuing until \(x_{\mathrm{A} i} \geq x_{\mathrm{Af}}\) Test the program supposing that the reaction is to be carried out at a temperature at which \(k=36.2 \mathrm{L} /(\mathrm{mol} \cdot \mathrm{min}),\) and that the other process variables have the following values: Use the program to calculate the required number of tanks and the final fractional conversion for the following values of the desired minimum final fractional conversion, \(x_{\mathrm{Af}}: 0.50,0.80,0.90,0.95\) 0.99, 0.999. Briefly describe the nature of the relationship between \(N\) and \(x_{\mathrm{Af}}\) and what probably happens to the process cost as the required final fractional conversion approaches \(1.0 .\) Hint: If you write a spreadsheet, it might appear in part as follows: (d) Suppose a 95\% conversion is desired. Use your program to determine how the required number of tanks varies as you increase (i) the rate constant, \(k ;\) (ii) the throughput, \(\dot{V} ;\) and (iii) the individual reactor volume, \(V\). Then briefly explain why the results you obtain make sense physically.

A drug (D) is produced in a three-stage extraction from the leaves of a tropical plant. About 1000 kg of leaf is required to produce 1 kg of the drug. The extraction solvent (S) is a mixture containing 16.5 wt\% ethanol (E) and the balance water (W). The following process is carried out to extract the drug and recover the solvent. 1\. A mixing tank is charged with \(3300 \mathrm{kg}\) of \(\mathrm{S}\) and \(620 \mathrm{kg}\) of leaf. The mixer contents are stirred for several hours, during which a portion of the drug contained in the leaf goes into solution. The contents of the mixer are then discharged through a filter. The liquid filtrate, which carries over roughly \(1 \%\) of the leaf fed to the mixer, is pumped to a holding tank, and the solid cake (spent leaf and entrained liquid) is sent to a second mixer. The entrained liquid has the same composition as the filtrate and a mass equal to \(15 \%\) of the mass of liquid charged to the mixer. The extracted drug has a negligible effect on the total mass and volume of the spent leaf and the filtrate. 2\. The second mixer is charged with the spent leaf from the first mixer and with the filtrate from the previous batch in the third mixer. The leaf is extracted for several more hours, and the contents of the mixer are then discharged to a second filter. The filtrate, which contains \(1 \%\) of the leaf fed to the second mixer, is pumped to the same holding tank that received the filtrate from the first mixer, and the solid cake- -spent leaf and entrained liquid - is sent to the third mixer. The entrained liquid mass is \(15 \%\) of the mass of liquid charged to the second mixer. 3\. The third mixer is charged with the spent leaf from the second mixer and with \(2720 \mathrm{kg}\) of solvent \(\mathrm{S}\). The mixer contents are filtered; the filtrate, which contains \(1 \%\) of the leaf fed to the third mixer, is recycled to the second mixer; and the solid cake is discarded. As before, the mass of the entrained liquid in the solid cake is \(15 \%\) of the mass of liquid charged to the mixer. 4\. The contents of the filtrate holding tank are filtered to remove the carried-over spent leaf, and the wet cake is pressed to recover entrained liquid, which is combined with the filtrate. A negligible amount of liquid remains in the wet cake. The filtrate, which contains \(\mathrm{D}, \mathrm{E},\) and \(\mathrm{W},\) is pumped to an extraction unit (another mixer). 5\. In the extraction unit, the alcohol-water-drug solution is contacted with another solvent (F), which is almost but not completely immiscible with ethanol and water. Essentially all of the drug (D) is extracted into the second solvent, from which it is eventually separated by a process of no concern in this problem. Some ethanol but no water is also contained in the extract. The solution from which the drug has been extracted (the raffinate) contains \(13.0 \mathrm{wt} \% \mathrm{E}, 1.5 \% \mathrm{F},\) and \(85.5 \%\) W. It is fed to a stripping column for recovery of the ethanol. 6\. The feeds to the stripping column are the solution just described and steam. The two streams are fed in a ratio such that the overhead product stream from the column contains \(20.0 \mathrm{wt} \% \mathrm{E}\) and \(2.6 \% \mathrm{F},\) and the bottom product stream contains \(1.3 \mathrm{wt} \% \mathrm{E}\) and the balance \(\mathrm{W}\). Draw and label a flowchart of the process, taking as a basis one batch of leaf processed. Then calculate (a) the masses of the components of the filtrate holding tank. (b) the masses of the components \(D\) and \(E\) in the extract stream leaving the extraction unit. (c) the mass of steam fed to the stripping column, and the masses of the column overhead and bottoms products.

A liquid mixture containing ethanol ( 55.0 wt\%) and the balance water enters a separation process unit at a rate of \(90.5 \mathrm{kg} / \mathrm{s}\). A technician draws samples of the two product streams leaving the separator and analyzes them with a gas chromatograph, obtaining values of 86.2 wt\% ethanol (Product Stream 1) and \(10.9 \%\) ethanol (Product Stream 2). The technician then reads a manometer attached to an orifice meter mounted in the pipe carrying Product Stream 1, converts the reading to a volumetric flow rate using a calibration curve, and converts that result to a mass flow rate using the average density of ethanol and water. The result is \(54.0 \mathrm{kg} / \mathrm{s}\). Finally, the technician calculates the mass flow rate of the second product stream using a material balance and reports the calculated product stream flow rates and compositions to you. You examine them, do some calculations, and reject them. (a) Draw and label a flow chart of the separation process. (b) Carry out the calculations that led you to reject the submitted results and explain how you knew the values were wrong. (c) List up to five possible reasons for the incorrect results. For each one, briefly state how you might determine whether it was in fact a cause of error and what you might do to correct it if it was.

Effluents from metal-finishing plants have the potential of discharging undesirable quantities of metals, such as cadmium, nickel, lead, manganese, and chromium, in forms that are detrimental to water and air quality. A local metal-finishing plant has identified a wastewater stream that contains 5.15 wt\% chromium (Cr) and devised the following approach to lowering risk and recovering the valuable metal. The wastewater stream is fed to a treatment unit that removes \(95 \%\) of the chromium in the feed and recycles it to the plant. The residual liquid stream leaving the treatment unit is sent to a waste lagoon. The treatment unit has a maximum capacity of 4500 kg wastewater/h. If wastewater leaves the finishing plant at a rate higher than the capacity of the treatment unit, the excess (anything above \(4500 \mathrm{kg} / \mathrm{h}\) ) bypasses the unit and combines with the residual liquid leaving the unit, and the combined stream goes to the waste lagoon. (a) Without assuming a basis of calculation, draw and label a flowchart of the process. (b) Wastewater leaves the finishing plant at a rate \(\dot{m}_{1}=6000 \mathrm{kg} / \mathrm{h}\). Calculate the flow rate of liquid to the waste lagoon, \(\dot{m}_{6}(\mathrm{kg} / \mathrm{h}),\) and the mass fraction of \(\mathrm{Cr}\) in this liquid, \(x_{6}(\mathrm{kg} \mathrm{Cr} / \mathrm{kg})\) (c) Calculate the flow rate of liquid to the waste lagoon and the mass fraction of Crin this liquid for \(\dot{m}_{1}\) varying from \(1000 \mathrm{kg} / \mathrm{h}\) to \(10,000 \mathrm{kg} / \mathrm{h}\) in \(1000 \mathrm{kg} / \mathrm{h}\) increments. Generate a plot of \(x_{6}\) versus \(\dot{m}_{1}\). (Suggestion: Use a spreadsheet for these calculations.) (d) The company has hired you as a consultant to help them determine whether or not to add capacity to the treatment unit to increase the recovery of chromium. What would you need to know to make this determination? (e) What concerns might need to be addressed regarding the waste lagoon?
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