We wish to explore various reactor setups for the transformation of A into R. The feed contains \(99 \%\) A, \(1 \%\) R; the desired product is to consist of \(10 \% \mathrm{A}, 90 \% \mathrm{R} .\) The transformation takes place by means of the elementary reaction \(A+R \rightarrow R+R\) with rate constant \(k=1\) liter/mol \(\cdot\) min. The concentration of active materials is \(C_{\mathrm{A} 0}+C_{\mathrm{R} 0}=C_{\mathrm{A}}+C_{\mathrm{R}}=C_{0}=1 \mathrm{mol} / / \mathrm{iter}\) throughout. What reactor holding time will yield a product in which \(C_{\mathrm{R}}=0.9 \mathrm{mol} /\) liter (a) in a plug flow reactor, (b) in a mixed flow reactor, and (c) in a minimum-size setup without recycle?

Step by step solution

01

Understand the Elementary Reaction

The given reaction is an elementary reaction represented by \(A+R \rightarrow R+R\), with a rate constant \(k=1\) liter/mol\(\cdot\)min. This implies the reaction rate is proportional to the concentrations of the reactants A and R.

02

Determine the Reaction Rate

The rate of reaction can be defined as \(-\frac{dC_A}{dt} = k \cdot C_A \cdot C_R\), where \(C_A\) and \(C_R\) are the concentrations of A and R, respectively, and \(k\) is the rate constant. Since \(C_A0 + C_R0 = C_A + C_R = C0 = 1\) mol/liter, we can substitute \(C_R\) with \(1 - C_A\) to get \(-\frac{dC_A}{dt} = k \cdot C_A \cdot (1 - C_A)\).

03

Calculate the Holding Time for a Plug Flow Reactor (PFR)

Integrate the reaction rate equation from the initial condition \(C_A(0)=0.99\) mol/liter to the final condition \(C_A(t)=0.10\) mol/liter to find the holding time \(t\) for the PFR. The differential equation to solve is \(-\frac{dC_A}{dt} = k \cdot C_A \cdot (1 - C_A)\) with respect to time \(t\). Integrate this equation to find the holding time \(t_PFR\).

04

Calculate the Holding Time for a Mixed Flow Reactor (MFR)

For an MFR, the exit concentration is the same as the concentration in the reactor, and the rate law becomes \(k \cdot C_A \cdot (1 - C_A) = \frac{C_A0 - C_A}{t}\). Given the exit concentration \(C_R=0.9\) mol/liter, we can find the corresponding \(C_A = 0.1\) mol/liter. Solve for \(t\) using the initial and final concentrations to find the holding time \(t_{MFR}\).

05

Determine the Minimum-Size Setup Without Recycle

For a minimum-size setup without recycle, we use a batch reactor where the entire volume of reactant is loaded at start. The total reaction time can be directly calculated like in PFR by integrating the rate law from initial to final concentration to get the batch reactor time \(t_{batch}\).

06

Perform Integration and Calculate Values

Perform the necessary integration for PFR and solve the algebraic equation for MFR to find the respective holding times. For the batch reactor, use the integrated expression from PFR as the calculation is similar.

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Key Concepts

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

Plug Flow Reactor

A plug flow reactor (PFR), also known as a tubular reactor, is a device where a chemical reaction proceeds as the reactants flow through a tube. The unique characteristic of a PFR is that the reactant mixture moves with a uniform velocity, meaning that elements of the fluid have no back-mixing and essentially

Mixed Flow Reactor

In contrast to a PFR, a mixed flow reactor (MFR) or continuous stirred-tank reactor (CSTR) operates with thorough mixing of its contents. This means the composition of the reaction mixture is uniform throughout the reactor.

When dealing with the transformation of chemical A into R in an MFR

Reaction Kinetics

Reaction kinetics involves the study of the rates of chemical reactions and how different conditions affect these rates. In our case, we look at the reaction where A transforms into R based on the elementary reaction with a given rate constant. Understanding the relationship between the concentration

Batch Reactor

A batch reactor is a closed system in which reactants are loaded, and then the reaction proceeds over time. No additional materials are added or removed until the reaction is deemed complete. Batch reactors are well-suited for reactions that need precise control over reaction time and conditions.

For our exercise,

Elementary Reaction

An elementary reaction is a chemical reaction that occurs in a single step and has a reaction rate that depends directly on the reactant’s concentrations raised to the power of their stoichiometric coefficients. In the textbook exercise, the reaction mentioned is elementary and follows a rate law where the rate of