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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

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.)

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]

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.)

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.)

A microreactor similar to the one shown in Figure P4-19 from the MIT group is used to produce phosgene in the gas phase. $$\begin{array}{c} \mathrm{CO}+\mathrm{Cl}_{2} \rightarrow \mathrm{COCl}_{2} \\ \mathrm{A}+\mathrm{B} \rightarrow \mathrm{C} \end{array}$$ The microreactor is \(20 \mathrm{mm}\) long. \(500 \mu \mathrm{m}\) in diameter, and packed with catalyst particles \(35 \mu \mathrm{m}\) in diameter. The entering pressure is \(830 \mathrm{kPa}(8.2 \mathrm{atm})\) and the entering flow to each microreactor is equimolar. The molar flow rate of \(\mathrm{CO}\) is \(2 \times 10^{-5} \mathrm{mol} / \mathrm{s}\) and the volumetric flow is \(2.83 \times 10^{-7} \mathrm{m}^{3} / \mathrm{s}\). The weight of catalyst in one microreactor: \(W=3.5 \times 10^{-6} \mathrm{kg}\). The reactor is kept isothermal at \(120^{\circ} \mathrm{C}\). Because the catalyst is also slightly different than the one in Figure \(\mathrm{P} 4-19,\) the rate law is different as well: $$-r_{A}^{\prime}=k_{A} C_{A} C_{B}$$ (a) Plot the molar flow rates \(F_{\mathrm{A}}, F_{\mathrm{B}}\), and \(F_{\mathrm{C}},\) the conversion \(X\), and pressure ratio \(y\) along the length of the reactor. (b) Calculate the number of microreactors in parallel to produce 10.000 kg/year phosgene. (c) Repeat part (a) for the case when the catalyst weight remains the same but the particle diameter is cut in half. If possible compare your answer with part (a) and describe what you find. noting anything unusual. (d) How would your answers to part (a) change if the reaction were reversible with \(K_{\mathrm{C}}=0.4 \mathrm{dm}^{3} / \mathrm{mol} ?\) Describe what you find. (e) What are the advantages and disadvantages of using an array of mi reactors over using one conventional packed bed reactor that provides same yield and conversion? (f) Write a question that involves critical thinking. and explain wh involves critical thinking. (g) Discuss what you learned from this problem and what you believe th the point of the problem. Additional information: \(\alpha=3.55 \times 10^{5} / \mathrm{kg}\) catalyst (based on properties of air and \(\phi=0.4\) ) \(k=0.004 \mathrm{m}^{6} / \mathrm{mol} \cdot \mathrm{s} \cdot \mathrm{kg}\) catalyst at \(120^{\circ} \mathrm{C}\) \(v_{0}=2.83 \cdot 10^{-7} \mathrm{m}^{3} / \mathrm{s}, \rho=7 \mathrm{kg} / \mathrm{m}^{3}, \mu=1.94 \cdot 10^{-5} \mathrm{kg} / \mathrm{m} \cdot \mathrm{s}\) \(A_{c}=1.96 \cdot 10^{-7} \mathrm{m}^{2}, G=10.1 \mathrm{kg} / \mathrm{m}^{2} \cdot \mathrm{s}\)
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