91Ó°ÊÓ

A liquid-phase chemical reaction \(\mathrm{A} \rightarrow \mathrm{B}\) takes place in a well-stirred tank. The concentration of \(\mathrm{A}\) in the feed is \(C_{\mathrm{A} 0}\left(\operatorname{mol} / \mathrm{m}^{3}\right),\) and that in the tank and outlet stream is \(C_{\mathrm{A}}\left(\mathrm{mol} / \mathrm{m}^{3}\right) .\) Neither concentration varies with time. The volume of the tank contents is \(V\left(\mathrm{m}^{3}\right)\) and the volumetric flow rate of the inlet and outlet streams is \(\dot{V}\left(\mathrm{m}^{3} / \mathrm{s}\right)\). The reaction rate (the rate at which \(\mathrm{A}\) is consumed by reaction in the tank) is given by the expression $$r(\text { mol } A \text { consumed } / \mathrm{s})=k V C_{\mathrm{A}}$$ (a) Is this process continuous, batch, or semibatch? Is it transient or steady-state? (b) What would you expect the reactant concentration \(C_{\mathrm{A}}\) to equal if \(k=0\) (no reaction)? What should it approach if \(k \rightarrow \infty\) (infinitely rapid reaction)? (c) Write a differential balance on \(A,\) stating which terms in the general balance equation (accumulation = input + generation - output - consumption) you discarded and why you discarded them. Use the balance to derive the following relation between the inlet and outlet reactant concentrations: $$C_{\mathrm{A}}=\frac{C_{\mathrm{A} 0}}{1+k V / \dot{V}}$$ Verify that this relation predicts the results in Part (b).

Step by step solution

01

Identify Process Type

From the problem statement, it can be inferred that the concentration in the tank does not vary with time and neither does it in the outlet and inlet streams. Therefore, this is a steady-state process. Moreover, the same flow rate for the inlet and outlet implies a flow in and out of the tank. This makes it a continuous process.

02

Reactant Concentration at Different Reaction Rates

If \(k=0\) meaning there is no reaction, no A is consumed. Hence, the inlet concentration would be the same as the tank and outlet concentrations. So, \(C_A=C_{A0}\). If \(k \rightarrow \infty\), the reaction would be infinitely fast. That means A would react out as soon as it enters the tank and the concentration in the tank would theoretically approach zero.

03

Write Differential Balance for A

For a steady-state process, accumulation term is zero. The generation term will be zero as A is not generated in the tank. Hence, the balance equation becomes: 0 = Input - Output - Consumption. Substituting the terms, we get \(0= C_{A0}\dot{V}-C_A\dot{V}-kVC_A \). Divide the equation by \( V\dot{V}\) to simplify it.

04

Derive the Relation

The simplified balance equation becomes: \(0= C_{A0} -C_A-kC_A\). By rearranging the terms, it yields \(C_A=\frac{C_{A0}}{1+kV/\dot{V}}\).

05

Verify the Derived Relation

For \(k=0\), substituting in derived equation gives \(C_A=C_{A0}\), which verifies for the condition of no reaction. For \(k \rightarrow \infty\), the denominator of the fraction in the derived relation becomes infinitely large making \(C_A=0\), which verifies for the condition of an infinitely fast reaction.

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Key Concepts

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

Reactor Design

Reactor design is a fundamental aspect of chemical reaction engineering that involves determining the most suitable type of reactor to conduct a specific chemical reaction efficiently and safely. There are various types of reactors, such as batch, continuous-stirred tank reactors (CSTR), plug flow reactors (PFR), and packed bed reactors, each suitable for different processes and reaction kinetics.

In our exercise, the reactor in question is a well-stirred tank, typically known as a CSTR, which is used for liquid-phase reactions. The design of such a reactor ensures complete mixing, implying uniform concentration and temperature throughout the reactor. The choice between a batch and continuous reactor depends on the desired operation mode, which affects the scalability, cost, and reaction control. In the given scenario, the reactor operates in a continuous mode as the feed is constantly supplied and products are removed, keeping the volume within the reactor constant over time. Understanding reactor design elements is crucial for optimizing reaction conditions, ensuring product quality, and scaling up processes in the industry.

Steady-State Process

A steady-state process is one where the variables (such as temperature, pressure, concentration) do not change with time. In the context of a chemical reactor, a steady-state operation means that the concentrations of reactants and products in the reactor remain constant over time, regardless of the changes that may occur in the feed. This implies that the rate of addition of reactants is balanced by the rate of removal of products and unreacted reactants.

In the example provided, the process is identified as steady-state due to the constant concentrations of reactant A in the feed and in the tank. This implies a balance between the input and output flows, which is characteristic of process industries striving for continuous production. Understanding steady-state conditions is vital for simplifying the calculations of reactor sizing and for analysis because it eliminates the time dependency, allowing for the use of algebraic equations instead of differential ones. The steady-state assumption is advantageous in industrial applications because it enables consistent product quality and simplifies control strategies.

Differential Balance

Differential balance, or material balance, is used to quantify how much of a reactant is present, being produced, consumed, or removed from a system over a period of time. The general balance equation includes terms for accumulation, input, generation, output, and consumption. However, in a steady-state process, the accumulation term is zero because the amount of material in the system isn't changing with time.

As for the given exercise, we are asked to consider a differential balance on species A. Since the process is at a steady-state and the reactor is well-mixed, there is no generation of A within the reactor. Therefore, the terms for accumulation and generation are discarded in the balance. What remains is to balance the input and output of A with its consumption by the reaction. Simplifying the equation and dividing by terms including flowrate and volume leads to a key relationship between the inlet concentration and the outlet concentration of A. This equation is invaluable for engineers as it succinctly relates kinetic parameters and process variables, enabling them to manipulate and control reactor systems optimally.