Trickle bed oxidation. Dilute aqueous ethanol (about \(2-3 \%\) ) is oxidized to acetic acid by the action of pure oxygen at 10 atm in a trickle bed reactor packed with palladium-alumina catalyst pellets and kept at \(30^{\circ} \mathrm{C}\). According to Sato et al., Proc. First Pacific Chem. Eng. Congress, Kyoto, p. 197,1972 the reaction proceeds as follows: $$\begin{array}{c} \mathrm{O}_{2}(g \rightarrow l)+\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}(l) \frac{\mathrm{on}}{\mathrm{catalyst}} \mathrm{CH}_{3} \mathrm{COOH}(l)+\mathrm{H}_{2} \mathrm{O} \\ -\frac{-a-\alpha-}{(\mathrm{A})} \end{array}$$ with rate $$-r_{\mathrm{A}}^{\prime}=k^{\prime} C_{\mathrm{A}}, \quad k^{\prime}=1.77 \times 10^{-5} \mathrm{m}^{3} / \mathrm{kg} \cdot \mathrm{s}$$ Find the fractional conversion of ethanol to acetic acid if gas and liquid are fed to the top of a reactor in the following system: Gas stream: \(\quad v_{g}=0.01 \mathrm{m}^{3} / \mathrm{s}, \quad H_{\mathrm{A}}=86000 \mathrm{Pa} \cdot \mathrm{m}^{3} / \mathrm{mol}\) Liquid stream: \(\quad v_{l}=2 \times 10^{-4} \mathrm{m}^{3} / \mathrm{s}, \quad C_{\mathrm{B} 0}=400 \mathrm{mol} / \mathrm{m}^{3}\) Reactor: \(\quad 5 \mathrm{m}\) high, \(\quad 0.1 \mathrm{m}^{2}\) cross section, \(\quad f_{s}=0.58\) \(\begin{array}{ll}\text { Catalyst: } & d_{p}=5 \mathrm{mm}, \quad \rho_{s}=1800 \mathrm{kg} / \mathrm{m}^{3} \\ & \mathscr{D}_{e}=4.16 \times 10^{-10} \mathrm{m}^{3} / \mathrm{m} \mathrm{cat} \cdot \mathrm{s}\end{array}\) \(\begin{array}{lll}\text { Kinetics: } & k_{\mathrm{A} g} a_{i}=3 \times 10^{-4} \mathrm{mol} / \mathrm{m}^{3} \cdot \mathrm{Pa} \cdot \mathrm{s}, & k_{\mathrm{A}} a_{i}=0.02 \mathrm{s}^{-1} \\ & k_{\mathrm{A} c}=3.86 \times 10^{-4} \mathrm{m} / \mathrm{s}\end{array}\)

Step by step solution

01

Understanding the problem

We need to find the fractional conversion of ethanol to acetic acid in a trickle bed reactor. The liquid ethanol and gaseous oxygen are reacting in the presence of a catalyst. The given rate expression is \(-r_{\mathrm{A}}^{'}=k^{'} C_{\mathrm{A}}\), where \(C_{\mathrm{A}}\) is the concentration of ethanol.

02

Calculate the molar flow rate of ethanol

First, calculate the molar flow rate of ethanol (component A) using the volumetric flow rate of the liquid and its initial concentration. \(F_{A0} = v_l \cdot C_{B0}\).

03

Calculate the superficial velocity of ethanol

The superficial velocity of ethanol (component A) can be calculated using the volumetric flow rate and the cross-section area of the reactor: \(u_{A0} = \frac{v_l}{A}\).

04

Determine the mole balance for ethanol

Apply the mole balance for ethanol in steady state, considering the trickle bed reactor operates like a plug flow reactor (PFR): \(\frac{dF_A}{dz} = r_{A}^{'} A\).

05

Solve for fractional conversion

Integrate the differential mole balance, assuming steady-state operation, to find the fractional conversion, \(X=\frac{F_{A0} - F_A}{F_{A0}}\), where \(F_A\) is the molar flow rate of ethanol at any position \(z\) in the reactor.

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Key Concepts

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

Chemical Reaction Engineering

Chemical reaction engineering is the branch that combines chemistry with mathematics and physics to study the design and function of chemical reactors. In the context of a trickle bed reactor, chemical reaction engineering involves understanding and applying the principles that govern the reaction kinetics, the contact between reactants, and the mass transfer inside the reactor.

In our particular exercise, the main task is to find the fractional conversion of ethanol to acetic acid. Chemists and engineers utilize these principles to optimize reactor design, ensuring reactions proceed efficiently under controlled conditions to yield the desired product.

Catalytic Oxidation

Catalytic oxidation is a process where a catalyst facilitates the oxidation of a substance. In the given exercise, palladium-alumina catalyst pellets are used to oxidize ethanol to acetic acid in the presence of oxygen. The catalyst increases the rate of the reaction without being consumed in the process.

Oxygen is typically poorly soluble in liquids; hence the design of the trickle bed reactor with an effective catalyst becomes crucial in ensuring a significant rate of reaction. The function of this catalyst is one of the most crucial points in the step-by-step solutions provided.

Fractional Conversion

Fractional conversion is a concept referring to the fraction of a reactant that has been converted into a product in a chemical reaction. It is defined as the amount of reactant consumed divided by the total amount initially present.

Mathematically, fractional conversion, usually denoted as 'X', is expressed as \( X = \frac{F_{A0} - F_A}{F_{A0}} \) where \( F_{A0} \) is the initial molar flow rate and \( F_A \) is the molar flow rate of the reactant at any point along the reactor. In our case, the objective is to calculate the fractional conversion of ethanol, which is essential in determining the efficiency of the catalytic process in the trickle bed reactor.

Reactor Design

The design of a reactor is pivotal to its performance and the efficiency of the conversion process. Factors that influence design include the type of reaction, the properties of the reactants and products, phase of the reactants, the desired conversion rate, safety, and cost-efficiency.

For trickle bed reactors, like the one described in the exercise, the design encompasses a packed bed through which the liquid (aqueous ethanol) trickles down while the gas (oxygen) moves up or concurrently downwards. The reactor dimensions, catalyst characteristics, and flow conditions all influence how effectively the reaction progresses. The specific design parameters provided in the task, such as height, cross-sectional area, and catalyst properties, all play a role in determining the fractional conversion of ethanol to acetic acid.

Rate Expression

Rate expressions are equations that provide a relationship between the reaction rate and the concentrations of reactants. They are an essential aspect of chemical kinetics and reactor design. The given rate expression for our trickle bed reactor scenario is \(-r_{A}^{'} = k^{'} C_{A}\), where \(-r_{A}^{'}\) is the rate of reaction per unit mass of catalyst, \(k^{'}\) is the rate constant, and \(C_{A}\) is the concentration of the reactant (ethanol).

This rate law implies that the reaction is first-order with respect to ethanol. Knowing the rate expression allows for the determination of the reaction rate under various reactor conditions and is crucial for solving the fractional conversion in the step-by-step solution.