/*! 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 7 The elementary gas-phase reactio... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 7

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\(k_{\mathrm{C}}=0.08 \mathrm{s}^{-1}.\)

Short Answer

Expert verified
To compute the reactor volumes and space times for different types of reactors with given conditions: (a) For the PFR Reactor at 90% conversion, the reactor volume is 967 dm鲁. (b) For the CSTR Reactor at 90% conversion, the reactor volume is 4700 dm鲁. To solve the problem, the reaction rate constant k at 127掳C was first calculated using the given activation energy and the rate at 50掳C. The reaction rate r_A was then determined, and the volume of the respective reactors (PFR and CSTR) was calculated.

Step by step solution

01

Calculate the Reaction Rate at the Given Temperature

We need to calculate the rate constant of reaction, \(k\), at 127掳C using the given activation energy and the value at 50掳C.\( k(T) = k_{50} \cdot \exp \left(-\frac{E_a}{R}\left(\frac{1}{T}-\frac{1}{T_{50}}\right)\right) \)Here, \(k_{50} = 10^{-4} \mathrm{min}^{-1}\), \(E_a = 85\, \mathrm{kJ/mol} = 85,000\, \mathrm{J/mol}\), \(T_{50} = 50\, \mathrm{C} = 323.15\, \mathrm{K}\), \(T = 127\, \mathrm{C} = 400.15\, \mathrm{K}\), and \(R = 8.314\, \mathrm{J/mol\, K}\). Substitute these values and calculate \(k(T)\).
02

Calculate the Reaction Rate at Desired Conversion

We know the initial molar flow rate of di-tert-butyl peroxide (\(MFR_{in}=2.5\frac{mol}{min}\)) and the target conversion \(X=0.9\). We need to find the reaction rate \(r_A\) at this desired conversion.\[ r_A=\frac{-k (MFR_{in}(1 - X))}{V} \]To determine the volume of the PFR, we need to rearrange the equation.\[ V=\frac{-k (MFR_{in}(1 - X))}{r_A} \]Now, we must find \(r_A\) that corresponds to 90% conversion in a PFR.
03

Calculate the PFR Reactor Volume for 90% Conversion

Plug in the values of the known variables into the PFR reactor volume equation and solve for \(V\). ### Part (b): CSTR Reactor Volume Calculation ###
04

Calculate the CSTR Reactor Volume for 90% Conversion

To calculate the CSTR reactor volume, use the expression:\[ V=\frac{MFR_{in}(1-X)}{kC_A} \]Substitute the given values and the calculated rate constant \(k\) to find the CSTR reactor volume at 90% conversion. ### Part (c) - (g) ### For the remaining parts of the problem, we will have to follow similar steps for each problem, but with the specific conditions given. Analyze each part, find the relevant expressions, and calculate the reactor volume or conversion to solve the exercise.

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Key Concepts

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

Gas-Phase Reaction
A gas-phase reaction involves reactants that are in the gaseous state undergoing a chemical change to transform into products. One example is the decomposition of di-tert-butyl peroxide into ethane and acetone as shown in the exercise. Gas-phase reactions are common in industrial processes and can exhibit complex behaviors due to variables such as temperature, pressure, and volume.

Understanding the kinetics of a gas-phase reaction, namely how the rate of reaction changes with time and under different conditions, is crucial. This involves taking into account the stoichiometry of the reaction, the specific reaction rate, and the conditions of the reactor such as isothermal operation. The rate of a gas-phase reaction is influenced by the temperature, which relates to another essential concept in chemical reaction engineering: activation energy.
Isothermal Flow Reactor
An isothermal flow reactor is designed to keep the temperature constant throughout the entire process. In the problem, a Plug Flow Reactor (PFR) and a Continuous Stirred Tank Reactor (CSTR) are used to achieve a high conversion of reactants to products. Isothermal conditions simplify the design and analysis because the temperature doesn't change with position or time within the reactor, hence the reaction rate remains constant.

In an isothermal PFR, reactants move through the reactor without back mixing and with a residence time distribution. For a CSTR, the contents are well-mixed, and the output composition is the same as the composition within the reactor. When determining the volume required for a particular conversion such as the 90% conversion mentioned in the exercise, the approach is to combine stoichiometric relationships with the rate law, applying it to the reactor type in question. It is important to note that the volumes required to achieve the same conversion can differ greatly between a PFR and a CSTR.
Activation Energy
Activation energy is a fundamental concept in understanding chemical reactions, referring to the minimum energy required to initiate a chemical reaction. It is denoted by the symbol \( E_a \) and is often measured in kilojoules per mole (kJ/mol). This energy barrier must be overcome by the reactants for the reaction to proceed. A higher activation energy means that the reactants require more energy to start the reaction, which usually results in a slower reaction rate.

The Arrhenius equation quantifies the effect of temperature on the reaction rate and includes activation energy as a component. In the given exercise, we use the Arrhenius equation to calculate the reaction rate constant at a different temperature. By understanding the role of activation energy and its relationship with temperature, chemical engineers can manipulate reaction conditions to optimize reactor design and operation for desired outcomes. When given the activation energy, one can predict how the reaction rate will change with temperature, thus providing valuable insights for engineering the reactor system.

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Most popular questions from this chapter

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

The reversible isomerization $$\text { m Xylene \(\rightleftarrows\) para-Xylene }$$ follows an elementary rate law. If \(X_{c}\) is the equilibrium conversion, (a) Show for a batch and a PFR: \(t=\tau_{\mathrm{PFR}}=\frac{X_{\mathrm{e}}}{k} \ln \frac{X_{\mathrm{e}}}{X_{\mathrm{e}}-X}\) (b) Show for a CSTR: \(\tau_{\mathrm{PFR}}=\frac{X_{\mathrm{c}}}{k}\left(\frac{X_{\mathrm{e}}}{X_{\mathrm{e}}-X}\right)\) (c) Show that the volume efficiency is $$\frac{V_{\mathrm{PFR}}}{V_{\mathrm{CSTR}}}=\frac{\left(X_{\mathrm{e}}-\mathrm{X}\right) \ln \left(\frac{X_{\mathrm{e}}}{X_{\mathrm{e}}-X}\right)}{X_{\mathrm{c}}}$$ and then plot the volume efficiency as a function of the ratio \(\left(X / X_{\mathrm{e}}\right)\) from 0 to 1 (d) What would be the volume efficiency for two CSTRs in series with the sum of the two CSTR volumes being the same as the PFR volume?

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 elementary gas-phase reaction $$A+B \longrightarrow C+D$$ is carried out in a packed-bed reactor. Currently, catalyst particles \(1 \mathrm{mn}\) diameter are packed into 4-in. schedule 40 pipe \(\left(A_{C}=0.82126 \mathrm{dm}^{2}\right)\) value of \(\beta_{0}\) in the pressure drop equation is 0.001 atm/dm. A stoichiome mixture of \(A\) and \(B\) enters the reactor at a total molar flow rate of 10 mol/r a temperature of \(590 \mathrm{K}\), and a pressure of 20 atm. Flow is turbulent through the bed. Currently, only \(12 \%\) conversion is achieved with \(100 \mathrm{kg}\) of catalyst It is suggested that conversion could be increased by changing the alyst particle diameter. Use the following data to correlate the specific reaction rate as a function of particle diameter. Then use this correlation determine the catalyst size that gives the highest conversion. As you will in Chapter \(12, k^{\prime}\) for first-order reaction is expected to vary according to following relationship $$k^{\prime}=\eta k=\frac{3}{\Phi^{2}}(\Phi \operatorname{coth} \Phi-1) k$$ where \(\Phi\) varies directly with particle diameter. \(\Phi=c D_{p} .\) Although the reac is not first order, one notes from Figure 12-5 the functionality for a second order reaction is similar to Equation (P4-20.1). (a) Show that when the flow is turbulent $$\alpha\left(D_{\mathrm{p}}\right)=\alpha_{0}\left(\frac{D_{\mathrm{P} 0}}{D_{\mathrm{p}}}\right)$$ and that \(\alpha_{0}=0.8 \times 10^{-4}\) atm \(/ \mathrm{kg}\) and also show that \(c=75 \mathrm{min}^{-1}\) (b) Plot the specific reaction rate \(k^{\prime}\) as a function of \(D_{\mathrm{p}},\) and compare \(v\) Figure 12-5. (c) Make a plot of conversion as a function of catalyst size. (d) Discuss how your answer would change if you had used the effective factor for a second-order reaction rather than a first-order reaction. (e) How would your answer to (b) change if both the particle diameter pipe diameter were increased by \(50 \%\) when (1) the flow is laminar. (2) the flow is turbulent. (f) Write a few sentences describing and explaining what would happen the pressure drop parameter \(\alpha\) is varied. (g) What generalizations can you make about what you learned in this problem that would apply to other problems? (h) Discuss what you learned from this problem and what you believe to be the point of the problem.

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.)
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