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Chapter 4: Problem 58

Carbon nanotubes (CNT) are among the most versatile building blocks in nanotechnology. These unique pure carbon materials resemble rolled-up sheets of graphite with diameters of several nanometers and lengths up to several micrometers. They are stronger than steel, have higher thermal conductivities than most known materials, and have electrical conductivities like that of copper but with higher currentcarrying capacity. Molecular transistors and biosensors are among their many applications. While most carbon nanotube research has been based on laboratory-scale synthesis, commercial applications involve large industrial-scale processes. In one such process, carbon monoxide saturated with an organo-metallic compound (iron penta-carbonyl) is decomposed at high temperature and pressure to form CNT, amorphous carbon, and CO_. Each "molecule" of CNT contains roughly 3000 carbon atoms. The reactions by which such molecules are formed are: In the process to be analyzed, a fresh feed of CO saturated with \(\mathrm{Fe}(\mathrm{CO})_{5}(\mathrm{v})\) contains \(19.2 \mathrm{wt} \%\) of the latter component. The feed is joined by a recycle stream of pure CO and fed to the reactor, where all of the iron penta-carbonyl decomposes. Based on laboratory data, \(20.0 \%\) of the CO fed to the reactor is converted, and the selectivity of CNT to amorphous carbon production is (9.00 kmol CNT/kmol C). The reactor effluent passes through a complex separation process that yields three product streams: one consists of solid \(\mathrm{CNT}, \mathrm{C},\) and \(\mathrm{Fe} ;\) a second is \(\mathrm{CO}_{2} ;\) and the third is the recycled \(\mathrm{CO}\). You wish to determine the flow rate of the fresh feed (SCM/h), the total CO_ generated in the process ( \(\mathrm{kg} / \mathrm{h}\) ), and the ratio (kmol CO recycled/kmol CO in fresh feed). (a) Take a basis of \(100 \mathrm{kmol}\) fresh feed. Draw and fully label a process flow chart and do degree-offreedom analyses for the overall process, the fresh-feed/recycle mixing point, the reactor, and the separation process. Base the analyses for reactive systems on atomic balances. (b) Write and solve overall balances, and then scale the process to calculate the flow rate (SCM/h) of fresh feed required to produce \(1000 \mathrm{kg} \mathrm{CNT} / \mathrm{h}\) and the mass flow rate of \(\mathrm{CO}_{2}\) that would be produced. (c) In your degree-of-freedom analysis of the reactor, you might have counted separate balances for C (atomic carbon) and O (atomic oxygen). In fact, those two balances are not independent, so one but not both of them should be counted. Revise your analysis if necessary, and then calculate the ratio (kmol CO recycled/kmol CO in fresh feed). (d) Prove that the atomic carbon and oxygen balances on the reactor are not independent equations.

Short Answer

Expert verified
The problem involves analyzing a process of synthesizing Carbon NanoTubes (CNT) from CO and \(\mathrm{Fe}(\mathrm{CO})_{5}\), drawing the relevant process flow, performing atomic and overall balances, and degree-of-freedom analyses. The process is also scaled up to determine the flow rate of CO_2 produced when 1000 kg/h of CNT is required. Additionally, it was shown that atomic carbon and oxygen balances are not independent in this case.

Step by step solution

01

Comprehensive Process Understanding

The problem involves the conversion of carbon monoxide (CO) and iron penta-carbonyl (\(\mathrm{Fe}(\mathrm{CO})_{5}\)) into carbon nanotubes (CNT), amorphous carbon (C), and iron (Fe). The basic understanding of this process is essential, including the feed constituents, conversion process, product streams, and effluents.
02

Drawing and Labelling the Process Flow Chart

A process flow diagram is to be created illustrating the CO and \(\mathrm{Fe}(\mathrm{CO})_{5}\) input, the reactor where the reaction takes place, the separation process, and the output streams which include CNT, C, CO_2 and recycled CO. Don't forget to mention the percent conversion, selectivity, and weight fractions where applicable.
03

Degree-of-Freedom Analysis

The degree of freedom is determined for the overall process, the fresh-feed/recycle mixing point, the reactor, and the separation process. This analysis involves determining the number of unknowns, the available balances and other equations, and subtracting the latter from the former.
04

Applying Material Balances

Material balances for carbon, oxygen, and iron in the reactor and overall process are written based on the reaction stoichiometry. For instance, carbon balance implies that the carbon entering the reactor with CO and \(\mathrm{Fe}(\mathrm{CO})_{5}\) equals the carbon leaving with CNT and CO_2, which can be written as \(CO_{in} + CNT_{out} = CO_{out} + \mathrm{Fe}(\mathrm{CO})_{5-in}\). Solution for these equations yields the flow rates of product and effluent streams.
05

Scaling up process

Using the results from the balance equations and given information, the process is scaled up to determine the flow rate of CO_2 produced when 1000 kg/h of CNT is required. This would demand proportional increase in the fresh feed flow rate as well.
06

Analysing Reactor's Degree of Freedom

The degree of freedom for the reactor is reanalyzed considering that the atomic carbon and oxygen balances are not independent (as the oxygen comes solely from the decomposition of CO). Therefore, only one of these balances should be included in the degree-of-freedom calculation.
07

Deriving Ratio of Recycled CO to CO in fresh feed

With the revised degree of freedom analysis, further calculation can be done to derive kmol CO recycled/kmol CO in fresh feed by using the balanced equations provided in step 4.
08

Independent Equation Analysis

An independent equation requires unique information that isn't represented in any other equation. To prove that atomic carbon and oxygen are not independent equations, it is enough to show that the information in the oxygen balance can be obtained from the carbon balance and vice versa, making them dependent. Hence, only one of them should be counted in the degree-of-freedom analysis.

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Key Concepts

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

Carbon Nanotubes
Carbon nanotubes (CNTs) represent a fascinating advancement in nanotechnology. Imagine sheets of graphite, intricately rolled into tiny tubes with diameters as narrow as a few nanometers and lengths extending to several micrometers. This unique structure gives CNTs extraordinary properties: they are exceptionally strong, exhibiting strength greater than steel. Their thermal conductivity surpasses many known materials, while their electrical conductivity rivals that of copper. This combination of properties makes them ideal for numerous applications like molecular transistors and biosensors.

However, producing CNTs on a commercial scale means moving from lab-bench experiments to large industrial processes. In these processes, specific reactions involve carbon monoxide and iron penta-carbonyl to produce not only CNTs but other forms of carbon, including amorphous carbon. Understanding these reactions and how CNTs form is key to efficiently designing processes that maximize CNT output while minimizing byproducts. As such, expertise in CNT production is critical for the development of new and existing technologies.
Process Flow Chart
Creating a process flow chart is like drafting a map of how substances move through a chemical process. It visually represents what enters and exits a process, as well as the pathways they take. For the case of carbon nanotube production, a flow chart would start with inputs such as carbon monoxide and iron penta-carbonyl, and track their transformation through a reactor.

This chart would include:
  • The reactor, where these compounds undergo decomposition.
  • The separation system, sorting solid CNTs and carbon from gases like carbon dioxide and remaining carbon monoxide.
  • Streams showing recycled and purged components.
Each part of the process must be labeled with pertinent information, such as feed rates, conversion percentages, and any material that gets recycled. A well-drafted flow chart aids engineers in understanding process efficiencies and where to apply improvements.

Through a comprehensive diagram, processes become more straightforward to manage, simplifying the complexity found in industrial-scale CNT production.
Material Balances
Material balances are fundamental to chemical engineering and ensure that all material entering a system is accounted for within the system and in the output. In the scenario of carbon nanotube synthesis, material balances involve tracking each atom or molecule through the process.

Key steps include:
  • Counting the carbon from CO and \( ext{Fe(CO)}_5 \) in the feed and comparing it to the carbon in CNTs and carbon dioxide in the products.
  • Assessing any byproducts, like amorphous carbon, and ensuring they match with the starting material.
This translates into equations derived from the stoichiometry of the reactions. For example, a material balance might be expressed as \( CO_{\text{in}} + CNT_{\text{out}} = CO_{\text{out}} + \left( ext{Fe(CO)}_5 \right)_{\text{in}} \). Solutions to these equations inform engineers of flow rates and compositions of various streams.

Proper material balancing ensures process efficiency and supports decision-making in design and operation.
Degree of Freedom Analysis
Degree of freedom analysis is a crucial tool in chemical process engineering that determines how many variables you can adjust independently without altering the system's equilibrium. It involves comparing the number of unknowns with the number of available equations. For a reactor where CNTs are produced, degree of freedom analysis ensures that the given process information is sufficient to solve for the unknowns.

Here's how you do it:
  • Identify all variables, such as flow rates or compositions of the inputs and outputs.
  • Write down all the balances, particularly for elements like carbon and oxygen.
  • Count the number of independent equations you have.
In this particular case, it's essential to recognize that not all balances (such as carbon and oxygen) are independent. The oxygen balance stems from a single source, complicating initial calculations. By re-evaluating which equations are dependent or redundant, one can accurately compute variables like the ratio of recycled CO to fresh feed CO.

Conducting a thorough degree of freedom analysis enhances our understanding of complex processes, ensuring more accurate design and optimization.

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Most popular questions from this chapter

A catalytic reactor is used to produce formaldehyde from methanol in the reaction $$\mathrm{CH}_{3} \mathrm{OH} \rightarrow \mathrm{HCHO}+\mathrm{H}_{2}$$ A single-pass conversion of \(60.0 \%\) is achieved in the reactor. The methanol in the reactor product is separated from the formaldehyde and hydrogen in a multiple-unit process. The production rate of formaldehyde is 900.0 kg/h. (a) Calculate the required feed rate of methanol to the process ( \(\mathrm{kmol} / \mathrm{h}\) ) if there is no recycle. (b) Suppose the unreacted methanol is recovered and recycled to the reactor and the single-pass conversion remains 60\%. Without doing any calculations, prove that you have enough information to determine the required fresh feed rate of methanol (kmol/h) and the rates (kmol/h) at which methanol enters and leaves the reactor. Then perform the calculations. (c) The single-pass conversion in the reactor, \(X_{\mathrm{sp}},\) affects the costs of the reactor \(\left(C_{\mathrm{r}}\right)\) and the separation process and recycle line \(\left(C_{\mathrm{s}}\right) .\) What effect would you expect an increased \(X_{\mathrm{sp}}\) would have on each of these costs for a fixed formaldehyde production rate? (Hint: To get a \(100 \%\) singlepass conversion you would need an infinitely large reactor, and lowering the single-pass conversion leads to a need to process greater amounts of fluid through both process units and the recycle line.) What would you expect a plot of \(\left(C_{\mathrm{r}}+C_{\mathrm{s}}\right)\) versus \(X_{\mathrm{sp}}\) to look like? What does the design specification \(X_{\mathrm{sp}}=60 \%\) probably represent?

A stream of humid air containing 1.50 mole \(\% \mathrm{H}_{2} \mathrm{O}(\mathrm{v})\) and the balance dry air is to be humidified to a water content of 10.0 mole\% \(\mathrm{H}_{2} \mathrm{O}\). For this purpose, liquid water is fed through a flowmeter and evaporated into the air stream. The flowmeter reading, \(R\), is \(95 .\) The only available calibration data for the flowmeter are two points scribbled on a sheet of paper, indicating that readings \(R=15\) and \(R=50\) correspond to flow rates \(\dot{V}=40.0 \mathrm{ft}^{3} / \mathrm{h}\) and \(\dot{V}=96.9 \mathrm{ft}^{3} / \mathrm{h},\) respectively. (a) Assuming that the process is working as intended, draw and label the flowchart, do the degree-offreedom analysis, and estimate the molar flow rate (lb-mole/h) of the humidified (outlet) air if (i) the volumetric flow rate is a linear function of \(R\) and (ii) the reading \(R\) is a linear function of \(\dot{V}^{0.5}\) (b) Suppose the outlet air is analyzed and found to contain only \(7 \%\) water instead of the desired \(10 \%\) List as many possible reasons as you can think of for the discrepancy, concentrating on assumptions made in the calculation of Part (a) that might be violated in the real process.

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.

\- An equimolar liquid mixture of benzene and toluene is separated into two product streams by distillation. A process flowchart and a somewhat oversimplified description of what happens in the process follow: Inside the column a liquid stream flows downward and a vapor stream rises. At each point in the column some of the liquid vaporizes and some of the vapor condenses. The vapor leaving the top of the column, which contains 97 mole\% benzene, is completely condensed and split into two equal fractions: one is taken off as the overhead product stream, and the other (the reflux) is recycled to the top of the column. The overhead product stream contains \(89.2 \%\) of the benzene fed to the column. The liquid leaving the bottom of the column is fed to a partial reboiler in which \(45 \%\) of it is vaporized. The vapor generated in the reboiler (the boilup) is recycled to become the rising vapor stream in the column, and the residual reboiler liquid is taken off as the bottom product stream. The compositions of the streams leaving the reboiler are governed by the relation $$\frac{y_{\mathrm{B}} /\left(1-y_{\mathrm{B}}\right)}{x_{\mathrm{B}} /\left(1-x_{\mathrm{B}}\right)}=2.25$$ where \(y_{\mathrm{B}}\) and \(x_{\mathrm{B}}\) are the mole fractions of benzene in the vapor and liquid streams, respectively. (a) Take a basis of 100 mol fed to the column. Draw and completely label a flowchart, and for each of four systems (overall process, column, condenser, and reboiler), do the degree-of-freedom analysis and identify a system with which the process analysis might appropriately begin (one with zero degrees of freedom). (b) Write in order the equations you would solve to determine all unknown variables on the flowchart, circling the variable for which you would solve in each equation. Do not do the calculations in this part. (c) Calculate the molar amounts of the overhead and bottoms products, the mole fraction of benzene in the bottoms product, and the percentage recovery of toluene in the bottoms product \((100 \times\) moles toluene in bottoms/mole toluene in feed).

A stream containing \(\mathrm{H}_{2} \mathrm{S}\) and inert gases and a second stream of pure \(\mathrm{SO}_{2}\) are fed to a sulfur recovery reactor, where the reaction $$2 \mathrm{H}_{2} \mathrm{S}+\mathrm{SO}_{2} \rightarrow 3 \mathrm{S}+2 \mathrm{H}_{2} \mathrm{O}$$ takes place. The feed rates are adjusted so that the ratio of \(\mathrm{H}_{2} \mathrm{S}\) to \(\mathrm{SO}_{2}\) in the combined feed is always stoichiometric. In the normal operation of the reactor the flow rate and composition of the \(\mathrm{H}_{2} \mathrm{S}\) feed stream both fluctuate. In the past, each time either variable changed the required \(\mathrm{SO}_{2}\) feed rate had to be reset by adjusting a valve in the feed line. A control system has been installed to automate this process. The \(\mathrm{H}_{2} \mathrm{S}\) feed stream passes through an electronic flowmeter that transmits a signal \(R_{\mathrm{f}}\) directly proportional to the molar flow rate of the stream, \(\dot{n}_{\mathrm{f}}\). When \(\dot{n}_{\mathrm{f}}=100 \mathrm{kmol} / \mathrm{h}\), the transmitted signal \(R_{\mathrm{f}}=15 \mathrm{mV}\). The mole fraction of \(\mathrm{H}_{2} \mathrm{S}\) in this stream is measured with a thermal conductivity detector, which transmits a signal \(R_{\mathrm{a}} .\) Analyzer calibration data are as follows: $$\begin{array}{|l|c|c|c|c|c|c|}\hline R_{\mathrm{a}}(\mathrm{mV}) & 0 & 25.4 & 42.8 & 58.0 & 71.9 & 85.1 \\ \hline x\left(\mathrm{mol} \mathrm{H}_{2} \mathrm{S} / \mathrm{mol}\right) & 0.00 & 0.20 & 0.40 & 0.60 &0.80 & 1.00 \\\\\hline\end{array}$$ The controller takes as input the transmitted values of \(R_{\mathrm{f}}\) and \(R_{\mathrm{a}}\) and calculates and transmits a voltage signal \(R_{\mathrm{c}}\) to a flow control valve in the \(\mathrm{SO}_{2}\) line, which opens and closes to an extent dependent on the value of \(R_{c} .\) A plot of the \(S O_{2}\) flow rate, \(\dot{n}_{c},\) versus \(R_{c}\) on rectangular coordinates is a straight line through the points \(\left(R_{c}=10.0 \mathrm{mV}, \dot{n}_{c}=25.0 \mathrm{kmol} / \mathrm{h}\right)\) and \(\left(R_{c}=25.0 \mathrm{mV}, \dot{n}_{c}=60.0 \mathrm{kmol} / \mathrm{h}\right)\) (a) Why would it be important to feed the reactants in stoichiometric proportion? (Hint: \(\mathrm{SO}_{2}\) and especially \(\mathrm{H}_{2} \mathrm{S}\) are serious pollutants.) What are several likely reasons for wanting to automate the \(\mathrm{SO}_{2}\) feed rate adjustment? (b) If the first stream contains 85.0 mole \(\% \mathrm{H}_{2} \mathrm{S}\) and enters the unit at a rate of \(\dot{n}_{\mathrm{f}}=3.00 \times 10^{2} \mathrm{kmol} / \mathrm{h}\) what must the value of \(\dot{n}_{c}\left(\mathrm{kmol} \mathrm{SO}_{2} / \mathrm{h}\right)\) be? (c) Fit a function to the \(\mathrm{H}_{2} \mathrm{S}\) analyzer calibration data to derive an expression for \(x\) as a function of \(R_{\mathrm{a}}\) Check the fit by plotting both the function and the calibration data on the same graph. (d) Derive a formula for \(R_{\mathrm{c}}\) from specified values of \(R_{\mathrm{f}}\) and \(R_{\mathrm{a}},\) using the result of Part (c) in the derivation. (This formula would be built into the controller.) Test the formula using the flow rate and composition data of Part (a). (e) The system has been installed and made operational, and at some point the concentration of \(\mathrm{H}_{2} \mathrm{S}\) in the feed stream suddenly changes. A sample of the blended gas is collected and analyzed a short time later and the mole ratio of \(\mathrm{H}_{2} \mathrm{S}\) to \(\mathrm{SO}_{2}\) is not the required 2: 1 . List as many possible reasons as you can think of for this apparent failure of the control system.
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