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Chapter 4: Problem 33

A CSTR with two impellers is modeled as three CSTRs in series.

Short Answer

Expert verified
A CSTR with two impellers can be modeled as three CSTRs connected in series. By analyzing the residence time of each reactor, the individual conversions can be determined using the first-order reaction equation: \( X_{i} = 1 - e^{ -k \tau_{i}}\). The overall conversion in the system is obtained by the equation: \(X_{total} = 1 - (1 - X_1)(1 - X_2)(1 - X_3)\).

Step by step solution

01

Understanding CSTRs

A Continuous Stirred Tank Reactor (CSTR) is a type of chemical reactor where the reactants are continuously added and products are continuously removed from the system. The reactor is well-mixed, which ensures that the composition, temperature, and other properties are uniform throughout the reactor.
02

Modeling a CSTR with two impellers as three CSTRs in series

Given that the CSTR system has two impellers, it can be modeled as three CSTRs connected in series. This means that the output of the first reactor feeds into the input of the second reactor, and the output of the second reactor feeds into the input of the third reactor. This representation is an approximation, but it helps to describe the mixing and residence time distribution within the system.
03

Analyze the residence time of each CSTR

The residence time of each reactor in the series can be analyzed to determine the overall performance of the system. The residence time is given by the equation: \( \tau = \dfrac{V}{Q} \) where \(\tau\) is the residence time, \(V\) is the volume of the reactor, and \(Q\) is the volumetric flow rate. By calculating the residence time of each CSTR in the series, we can estimate the total residence time of the system and the conversion of reactants to products.
04

Calculating conversion in each CSTR

The conversion of reactants to products can be calculated using the following expression for a first-order reaction: \(X = 1 - e^{-kt} \) where X is the conversion, k is the reaction rate constant, and t is the reaction time. Since each CSTR is operating continuously, the reaction time can be approximated by the residence time (\(\tau\)) of that reactor. Therefore, the conversion in each reactor can be calculated as: \( X_{i} = 1 - e^{ -k \tau_{i}}\) where \(X_{i}\) is the conversion in each reactor and \(\tau_{i}\) is the residence time of each reactor.
05

Overall conversion in the system

The overall conversion of the system can be determined by analyzing the conversion in each reactor and how they are connected in series. To find the overall conversion, we can use the following equation: \(X_{total} = 1 - (1 - X_1)(1 - X_2)(1 - X_3)\) This equation takes into account that each reactor contributes to the overall conversion, and the fact that they are connected in series makes the overall conversion a product of their individual conversions. By following these steps, one can understand the concept of a CSTR system with two impellers modeled as three reactors in series and analyze the performance of this system by calculating residence times, individual reactor conversions, and overall conversion.

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Key Concepts

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

Chemical Reactor Design
Chemical reactor design is crucial in determining how efficiently chemical reactions are carried out within a reactor. A typical design involves choosing the right type and configuration of reactors to achieve the intended chemical transformation efficiently and safely. Continuous Stirred Tank Reactors (CSTRs) are popular due to their simplicity and effectiveness in maintaining uniform conditions within the vessel.

A CSTR fosters a well-mixed environment which helps in maintaining a consistent reaction environment. They are commonly used in industries where large-scale continuous production is required, such as in petrochemical and pharmaceutical industries.
  • Uniform mixing achieves homogeneity in concentration and temperature, which is vital for predictable reaction kinetics.
  • Their design is also flexible, allowing multiple CSTRs to be connected in series for enhanced operational control.
  • The series configuration helps mimic plug flow reactor characteristics, achieving higher conversion levels and more efficient use of reactants.
Reaction Kinetics
Reaction kinetics involves studying the speed of chemical reactions and the factors influencing them, such as temperature, concentration, and catalysts. In a CSTR, this is about optimizing these factors to maximize reaction rates and therefore conversion.

The rate of chemical reactions is often modeled by equations like the first-order reaction kinetics equation, where the conversion is defined by the formula: \( X = 1 - e^{-kt} \). Understanding these kinetics forms the backbone of chemical reactor designs like CSTRs.

  • Kinetic studies help in determining appropriate operating conditions to achieve the desired conversion.
  • By calculating reaction rates, engineers can predict the performance and efficiency of reactors, allowing for design adjustments.
  • Kinetics also aids in scaling up laboratory results to industrial-scale production.
Residence Time Distribution
Residence time distribution (RTD) provides insights on how long reactants spend in the reactor, influencing the extent of reaction. This is key when considering CSTRs in series, as each unit's RTD affects the overall system performance.

In a CSTR, residence time is depicted by the equation \( \tau = \frac{V}{Q} \), where \( V \) is the reactor volume and \( Q \) is the volumetric flow rate. This metric helps assess how effectively reactants are being utilized and converted to products.

  • RTD analysis is crucial for predicting the conversion efficiency, particularly in multicombination CSTR setups.
  • Understanding RTD helps in diagnosing operational issues like bypassing or dead zones in the reactor.
  • Efficient RTD indicates proper mixing and interaction of reactants within the reactor.
Series Reactors Modeling
Series reactors modeling is a strategy for enhancing reaction rates and improving conversion efficiency by connecting multiple reactors in a sequence. Modeling a CSTR with two impellers as three CSTRs in series is an example of this approach.

The idea behind this model is to mimic the behavior of a plug flow reactor while taking advantage of the simplicity of CSTR operations. By breaking down the reaction process into stages, the overall conversion can be maximized, as each reactor pushes the reaction closer to completion.

  • Series configuration allows for better control over reaction stages, leading to improved selectivity of products.
  • It reduces the variation seen in product output, ensuring consistent quality.
  • This model is particularly useful in processes requiring stringent conditions for optimal reaction quality.

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Most popular questions from this chapter

Compound A undergoes a reversible isomerization reaction, \(\mathrm{A} \rightleftarrows \mathrm{B}\). over a supported metal catalyst. Under pertinent conditions, A and B are liquid, miscible, and of nearly identical density; the equilibrium constant for the reaction (in concentration units) is 5.8. In a fixed-bed isothermal fow reactor in which backmixing is negligible (i.e... plug flow), a feed of pure \(A\) undergoes a net conversion to \(\mathrm{B}\) of \(55 \% .\) The reaction is elementary. If a second, identical flow reactor at the same temperature is placed downstream from the first, what overall conversion of A would you expect if: (a) The reactors are directly connected in series? (Ans.: \(X=0.74\) ) (b) The products from the first reactor are separated by appropriate processing and only the unconverted \(A\) is fed to the second reactor? (From California Professional Engineers Exam.)

It is desired to carry out the gaseous reaction \(A \longrightarrow B\) in an existing tubular reactor consisting of 50 parallel tubes 40 ft long with a 0.75-in. inside diameter. Bench-scale experiments have given the reaction rate constant for this first-order reaction as \(0.00152 \mathrm{s}^{-1}\) at \(200^{\circ} \mathrm{F}\) and \(0.0740 \mathrm{s}^{-1}\) at \(300^{\circ} \mathrm{F}\). At what temperature should the reactor be operated to give a conversion of \(\mathrm{A}\) of \(80 \%\) with a feed rate of \(500 \mathrm{lb} / \mathrm{h}\) of pure \(\mathrm{A}\) and an operating pressure of 100 psig? A has a molecular weight of \(73 .\) Departures from perfect gas behavior may be neglected, and the reverse reaction is insignificant at these conditions. (Ans.: \(T=275^{\circ} \mathrm{F}\) ) (From California Professional Engineers Exam.)

Dibutyl phthalate (DBP), a plasticizer, has a potential market of 12 million Ib/yr (AIChE Student Contest Problem) and is to be produced by reaction of n-butanol with monobutyl phthalate (MBP). The reaction follows an elementary rate law and is catalyzed by \(\mathrm{H}_{2} \mathrm{SO}_{4}\) (Figure \(\mathrm{P} 4-6\) ). A stream containing MBP and butanol is to be mixed with the \(\mathrm{H}_{2} \mathrm{SO}_{4}\) catalyst immediately before the stream enters the reactor. The concentration of MBP in the stream entering the reactor is \(0.2 \mathrm{tb} \mathrm{mol} / \mathrm{ft}^{3}\), and the molar feed rate of butanol is five times that of MBP. The specific reaction rate at \(100^{\circ} \mathrm{F}\) is \(1.2 \mathrm{ft}^{3} / \mathrm{lb}\) mol \(\cdot \mathrm{h}\) There is a 1000 -gallon CSTR and associated peripheral equipment available for use on this project for 30 days a year (operating 24 h/day). (a) Determine the exit conversion in the available 1000 -gallon reactor if you were to produce \(33 \%\) of the share (i.e., 4 million \(\mathrm{Ib} / \mathrm{yr}\) ) of the predicted market. (Ans.: \(X=0.33\) ) (b) How might you increase the conversion for the same \(F_{\mathrm{AO}} ?\) For example, what conversion would be achieved if a second 1000 -gal CSTR were placed either in series or in parallel with the CSTR? [\(X_{2}=0.55\) (series)] \right. (c) For the same temperature as part (a), what CSTR volume would be necessary to achieve a conversion of \(85 \%\) for a molar feed rate of \(\mathrm{MBP}\) of 1 Ib mol/min? (d) If possible. calculate the tubular reactor volume necessary to achieve \(85 \%\) conversion. when the reactor is oblong rather than cylindrical, with a major-to-minor axis ratio of \(1.3: 1.0 .\) There are no radial gradients in either concentration or velocity. If it is not possible to calculate \(\mathrm{V}_{\mathrm{PRF}}\) explain. (e) How would your results for parts (a) and (b) change if the temperature were raised to \(150^{\circ} \mathrm{F}\) where \(k\) is now \(5.0 \mathrm{ft}^{3} / \mathrm{lb}\) mol \(\cdot \mathrm{h}\) but the reaction is reversible with \(K_{C}=0.3 ?\) (f) Keeping in mind the times given in Table 4-1 for filling, and other operations, how many 1000 -gallon reactors operated in the batch mode would be necessary to meet the required production of 4 million pounds in a 30-day period? Estimate the cost of the reactors in the system. Note: Present in the feed stream may be some trace impurities, which you may lump as hexanol. The activation energy is believed to be somewhere around 25 kcal/mol. Hint: Plot number of reactors as a function of conversion. ( \(A n\) Ans.: 5 reactors) (g) What generalizations can you make about what you learned in this problem that would apply to other problems? (h) Write a question that requires critical thinking and then explain why your question requires critical thinking. [Hint: See Preface. Section B.2]

The gaseous reaction \(A \longrightarrow B\) has a unimolecular reaction rate constant of \(0.0015 \mathrm{min}^{-1}\) at \(80^{\circ} \mathrm{F}\). This reaction is to be carried out in parallel tubes \(10 \mathrm{ft}\) long and 1 in. inside diameter under a pressure of 132 psig at \(260^{\circ} \mathrm{F}\). A production rate of \(1000 \mathrm{lb} / \mathrm{h}\) of \(\mathrm{B}\) is required. Assuming an activation energy of 25,000 cal/mol. how many tubes are needed if the conversion of \(A\) is to be \(90 \% ?\) Assume perfect gas laws. A and \(\mathrm{B}\) each have molecular weights of 58 . (From California Professional Engineers Exam.)

The elementary gas-phase reaction $$\left(\mathrm{CH}_{3}\right)_{3} \operatorname{COOC}\left(\mathrm{CH}_{3}\right)_{3} \rightarrow \mathrm{C}_{2} \mathrm{H}_{6}+2 \mathrm{CH}_{3} \mathrm{COCH}_{3}$$ is carried out isothermally in a flow reactor with no pressure drop. The specific reaction rate at \(50^{\circ} \mathrm{C}\) is \(10^{-4} \mathrm{min}^{-1}\) (from pericosity data) and the activation energy is \(85 \mathrm{kJ} / \mathrm{mol}\). Pure di-tert-butyl peroxide enters the reactor at \(10 \mathrm{atm}\) and \(127^{\circ} \mathrm{C}\) and a molar flow rate of \(2.5 \mathrm{mol} / \mathrm{min}\). Calculate the reactor volume and space time to achieve \(90 \%\) conversion in: (a) a PFR (Ans.: 967 \(\mathrm{dm}^{3}\)) (b) a CSTR (Ans.: 4700 \(\mathrm{dm}^{3}\)) (c) Pressure drop. Plot \(X\). y, as a function of the PFR volume when \(\alpha=0.001\) \(\mathrm{dm}^{-3} .\) What are \(X .\) and \(y\) at \(V=500 \mathrm{dm}^{3} ?\) (d) Write a question that requires critical thinking. and explain why it involves critical thinking. (e) If this reaction is to be carried out isothermally at \(127^{\circ} \mathrm{C}\) and an initial pressure of 10 atm in a constant-volume batch mode with \(90 \%\) conversion. what reactor size and cost would be required to process \((2.5 \mathrm{mol} / \mathrm{min}\) \(\times 60 \min / \mathrm{h} \times 24 \text { h/day }) 3600\) mol of di-tert-butyl peroxide per day? (Hint: Recall Table 4-1.) (f) Assume that the reaction is reversible with \(K_{C}=0.025 \mathrm{mol}^{2} / \mathrm{dm}^{6}\). and calculate the equilibrium conversion; then redo (a) through (c) to achieve a conversion that is \(90 \%\) of the equilibrium conversion. (g) Membrane reactor. Repeat Part (f) for the case when \(\mathrm{C}_{2} \mathrm{H}_{6}\) flows out through the sides of the reactor and the transport coefficient is\(k_{\mathrm{C}}=0.08 \mathrm{s}^{-1}.\)
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