Problem 4 - Chemical Reaction Engineering

Question

Aqueous \(\mathrm{A}\left(C_{\mathrm{A} 0}=1 \text { mol/liter }\right)\) with physical properties close to water \(\left(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}, \mathscr{D}=10^{-9} \mathrm{m}^{2} / \mathrm{s}\right)\) reacts by a first-order homogeneous reaction \(\left(\mathrm{A} \rightarrow \mathrm{R}, k=0.2 \mathrm{s}^{-1}\right)\) as it flows at \(100 \mathrm{mm} / \mathrm{s}\) through a tubular reactor \(\left(d_{t}=50 \mathrm{mm}, L=5 \mathrm{m}\right) .\) Find the conversion of \(\mathrm{A}\) in the fluid leaving this reactor.

Step-by-Step Solution

Answer

After calculating residence time \(\tau \approx 50 \text{s}\), the exit concentration of A \([A]_e\) is found using \(\ln[A]_e = -k\tau + \ln[A_0]\), leading to conversion X.

Step 1: Understanding the Homogeneous First-Order Reaction

A first-order reaction follows the rate law: Rate = k[A], where k is the rate constant, and [A] is the concentration of reactant A. The conversion, X, can be defined as the fraction of A that has reacted. Therefore, the change of concentration of A over time can be described by the equation: \[\frac{d[A]}{dt} = -k[A]\]

Step 2: Simplifying the Differential Rate Equation

Integrate the differential rate equation \[\frac{d[A]}{dt} = -k[A]\] to find the concentration of A at any time t. The integrated form is \[\ln[A] = -kt + \ln[A_0]\] where \([A_0]\) is the initial concentration of A, and \([A]\) is the concentration at time t.

Step 3: Calculating the Residence Time

Residence time (\(\tau\)) is the time it takes for a fluid element to pass through the reactor. Calculate it using the formula: \[\tau = \frac{L}{v}\] where L is the length of the reactor and v is the linear flow velocity.

Step 4: Determine the Concentration of A at the Reactor Exit

With the residence time, the exit concentration of A (\([A]_e\)) after reaction time \(\tau\) can be determined by applying the residence time to the integrated rate equation: \[\ln[A]_e = -k\tau + \ln[A_0]\]

Step 5: Calculate the Conversion of A

The conversion, X, can be calculated from the initial and exit concentrations using: \[X = \frac{[A]_0 - [A]_e}{[A]_0}\]

Step 6: Plug Values into the Equations to Find Conversion X

Use the given values \(k = 0.2 \text{s}^{-1}\), \(d_t = 50 \text{mm}\), \(L = 5 \text{m}\), \(v = 100 \text{mm/s}\), and \([A_0] = 1 \text{mol/liter}\) to find the value of \(\tau\), and then use \(\tau\) to find \(\ln[A]_e\). Finally, use the exit concentration to find the conversion X.

Key Concepts

Chemical Reaction Engineering

Chemical Reaction Engineering (CRE) is the field that quantitatively analyzes chemical reactions to design and operate reactor systems effectively. CRE looks closely at finding the optimal conditions to enhance the yield of products while minimizing costs and ensuring safety. It intertwines with kinetics, the study of the rates of chemical processes, to understand how factors like temperature, pressure, and concentration impact the transformation of reactants into products.

A core aspect of CRE is the understanding of reaction mechanisms and rate laws, such as the first-order reaction in our exercise. Knowing that the rate of reaction is directly proportional to the reactant concentration in a first-order process helps engineers to predict how fast a reactant will be consumed and design reactors accordingly. Effective design and operation of reactors, such as the tubular reactor from our example, require a robust knowledge of these principles for optimization and scale-up from laboratory to industrial production.

The equation given:

• \( \mathrm{Rate} = k[A] \)

shows a direct connection between rate and concentration, indicating its first-order nature. This foundational equation also sets the stage for more complex calculations essential for designing and optimizing reactors.

Conversion of Reactants

Conversion of reactants refers to the extent to which reactants transform into products in a chemical process. It is a critical parameter that indicates the efficiency of a chemical reactor. For students and engineers, understanding conversion is pivotal for designing reactors and evaluating their performance.

Mathematically, conversion is expressed as:

• \( X = \frac{[A]_0 - [A]_e}{[A]_0} \)

where

• \( X \)

is the conversion factor,

• \( [A]_0 \)

is the initial concentration of reactants, and

• \( [A]_e \)

is the concentration of reactants at the exit of the reactor. High conversion rates usually mean more efficient reactor systems, but this must be balanced against the cost and feasibility of achieving such conversions.

In the context of our exercise, determining conversion is essential for measuring how successful the tubular reactor is at transforming Reactant A into Product R. By calculating the conversion, engineers can make informed decisions on process improvements and optimizations.

Residence Time Calculation

Residence time, represented by the Greek letter tau (\( \tau \)), is a key concept in CRE, denoting the average time a reactant particle spends in a reactor. It is a crucial factor that influences conversion, as it directly affects the amount of time reactants are exposed to reaction conditions. The longer the residence time, the more opportunity for reactants to convert to products, up to a certain limit depending on the reaction kinetics.

The formula used to calculate residence time is:

• \( \tau = \frac{L}{v} \)

where

• \( L \)

is the length of the reactor and

• \( v \)

is the linear velocity of fluid flow through the reactor. For a tubular reactor like the one in our exercise, residence time aids in predicting the final conversion of reactants at the reactor outlet.

By combining the concepts of residence time and reaction kinetics, as shown in the solution provided:

• \( \ln[A]_e = -k\tau + \ln[A_0] \)

students can understand the interplay between the physical dimensions and flow characteristics of the reactor and the chemical reaction that takes place within its confines. This knowledge culminates in the ability to design reactors that optimize production yields and efficiency.