91Ó°ÊÓ

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.

Step by step solution

01

Material balance for species A

The first part involves forming an overall material balance for species A over the ith tank. Then, taking into account the rate of reaction, this balance will result in the expression: \(V\frac{dC_{A}}{dt} = V\dot{V}(C_{A_{(i-1)}} - C_{A}) - V.k.C_{A}.C_{B}\) where \(C_{A_{(i-1)}}\) and \(C_{A}\) are the concentrations at the inlet and outlet of the ith reactor. Then, in steady state \(dC_{A}/dt = 0\), leading to \(\tau\Delta C_{A_{i}}=-\tau k C_{A_{i}} C_{B_{i}}\)

02

Material balance on species B

Next, we follow the same procedure as in step one, but now for species B. This results in another equation. We subtract the two equations and get \(\frac{\Delta C_{A_i}}{\Delta C_{B_i}} = \frac{C_{A_{0_i}} - C_{A_i}}{C_{B_{0_i}} - C_{B_i}}\)

03

Formulate equation for concentration

Using previous expressions, we can prove that \(\frac{C_{A_{0_i}}}{C_{A_i}} = 1 + k\tau (C_{B_{0_i}} - C_{A_{0_i}})\) Then, rearranging the terms and introducing some constants, we obtain an equation of the form: \(C_{A_{i}} = \frac{\alpha}{\beta + \gamma C_{A_{i-1}}}\)

04

Coding a program

The program will take input values of k, \(\dot{V}\), V, \(C_{A_{0}}\), \(C_{B_{0}}\), and \(x_{A_f}\) Use the equation we derived in the previous step to calculate the concentration and the corresponding fractional conversion. We then repeat this process until the desired fractional conversion is reached. This will provide us the number of reactors required.

05

Relationship between N and \(x_{A_f}\)

By running the program for different desired conversion \(x_{A_f}\), we find out that N increases as \(x_{A_f}\) approaches 1. This is because as the desired conversion increases, more tanks are needed to achieve the higher conversion.

06

Analysis of the Influence of parameters

A final simulation is made to evaluate how the number of tanks changes when the rate constant, the flowrate and the reactor volume are varied. These results are then compared and physically interpreted.

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Key Concepts

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

Material Balance

Material balance is a fundamental principle in chemical engineering that involves accounting for the flow of materials into, through, and out of a system. It is rooted in the law of conservation of mass, which states that mass cannot be created or destroyed. In chemical reaction engineering, a material balance equation for a particular component (such as a reactant or a product) in a reaction vessel will typically have the form of an input minus an output plus a generation term minus a consumption term.

When computing a material balance for a reactor, we consider the concentration of reactants entering the system, any changes within the reactor, and the final concentration of products leaving the system. For a continuous flow reactor, such as the stirred-tank reactors in the textbook example, the material balance becomes important to establish the relationship between feed concentrations, reaction kinetics, and the reactor volume. In the case of the ethyl acetate reaction with sodium hydroxide, the relevant material balance for component A resulted in an equation that, at steady state where the accumulation term is zero, expressed the change in concentration of A as a function of residence time and reaction rate with component B. Understanding and correctly applying material balances ensures the effectiveness of reactor design and optimization in chemical processes.

Reaction Kinetics

Reaction kinetics examines the rates at which chemical reactions progress and determines the factors that influence these rates, such as concentration, temperature, and catalyst presence. The rate of a reaction can be described by a rate equation, which provides a linkage between the concentration of reactants and the speed at which they convert to products.

In the context of the textbook problem, the rate of the reaction between ethyl acetate (A) and sodium hydroxide (B) is given by a rate constant (\( k \)) multiplied by the product of the concentrations of the two reactants (\( k \times C_A \times C_B \)). Reaction kinetics is deeply intertwined with reactor design since the kinetics dictate the size and type of reactor needed to achieve a certain product yield or conversion. An understanding of kinetics is crucial to calculate the amount of reactant that will be converted in a given reactor setup, as was necessary in step 3 of the problem's solution, and informs the strategies to improve the reaction's efficiency. For example, increasing the temperature or adding a catalyst might enhance the rate constant (\( k \)), thus increasing reaction rates and potentially reducing the number of reactors needed in a series to reach a desired conversion.

Reactor Design

Reactor design is the process of planning and constructing a reactor to carry out chemical reactions efficiently and safely. Reactor design considerations include the type of reaction, the phase of the reactants and products, the temperature and pressure operating conditions, and the desired conversion or yield. Various types of reactors, such as batch, plug flow, and continuous stirred-tank reactors (CSTRs), are used depending on the specific application and economic aspects.

In our scenario, a series of CSTRs is used, a common choice for liquid-phase reactions that benefit from constant agitation and uniformity in composition and temperature. The design of these reactors relies heavily on the previously discussed material balances and reaction kinetics. By utilizing the balanced equations and kinetic information, the designer can determine the optimal size, the number of reactors, and the configuration that would achieve the desired conversion. Optimization efforts may also consider cost, space, and energy efficiency. In the given exercise, the task was to use the derived equations to compute the number of stirred-tank reactors needed to achieve varying fractional conversions, acknowledging the relationship between reactor volume, throughput, rate constant, and the number of reactors required to reach a specific performance.

Steady State Analysis

Steady state analysis in chemical reaction engineering focuses on conditions where the variables that describe the system behavior, such as temperatures, pressures, and concentrations, do not change with time. This assumption simplifies the analysis of reactors by allowing us to set the derivative of the concentration with respect to time to zero, as was done in step 1 of the solution provided for the textbook problem. It essentially means that the rate of material entering a reactor is equal to the rate of material leaving, plus any reaction occurring within the reactor.

Steady state analysis is important when considering the operation of a series of reactors because it allows for the calculation of concentrations and conversions within each reactor as if each were operating independently. This method was applied to calculate the concentrations of the reactants and the extent of conversion in each subsequent tank. The analysis highlights that as the required conversion approaches 100%, an increasing number of reactors is needed, which, in turn, drives up the process cost. It's a critical consideration in the economic feasibility of a chemical process and underscores the importance of designing a system that optimally balances between conversion efficiency and cost.