Question

The irreversible gas-phase reaction $$A \stackrel{\text { cealyst }}{\longrightarrow} \text { B }$$ is carried out adiabatically over a packed bed of solid catalyst particles. The reaction is first order in the concentration of \(A\) on the catalyst surface: $$-r_{\mathrm{A} s}^{\prime}=k^{\prime} C_{\mathrm{A}}$$ The feed consists of \(50 \%\) (mole) \(A\) and \(50 \%\) inerts and enters the bed at a temperature of \(300 \mathrm{K}\). The entering volumetric flow rate is \(10 \mathrm{dm}^{3} / \mathrm{s}\) (i.e... \(10,000 \mathrm{cm}^{3} / \mathrm{s}\) ). The relationship between the Sherwood number and the Reynolds number is $$S h=100 \mathrm{Re}^{1 / 2}$$ As a first approximation, one may neglect pressure drop. The entering concentration of \(A\) is \(1.0 \mathrm{M}\). Calculate the catalyst weight necessary to achieve \(60 \%\) conversion of \(\mathrm{A}\) for (a) Isothermal operation. (b) Adiabatic operation. (c) What generalizations can you make after comparing parts (a) and (b)? Additional information: Kinematic viscosity: \(\mu / \rho=0.02 \mathrm{cm}^{2} / \mathrm{s}\) Particle diameter: \(d_{p}=0.1 \mathrm{cm}\) Superficial velocity: \(U=10 \mathrm{cm} / \mathrm{s}\) Catalyst surface area/mass of catalyst bed: \(a=60 \mathrm{cm}^{2} / \mathrm{g}\) cat. Diffusivity of \(\mathrm{A}: D_{e}=10^{-2} \mathrm{cm}^{2} / \mathrm{s}\) Heat of reaction: \(\Delta H_{\mathrm{Rx}}=-10,000 \mathrm{cal} / \mathrm{g}\) mol \(\mathrm{A}\) Heat capacities: $$\begin{aligned} C_{p A}=C_{p B} &=25 \mathrm{cal} / \mathrm{g} \mathrm{mol} \cdot \mathrm{K} \\\ C_{p S}(\text { solvent }) &=75 \mathrm{cal} / \mathrm{g} \mathrm{mol} \cdot \mathrm{K} \end{aligned}$$ $$k^{\prime}(300 \mathrm{K})=0.01 \mathrm{cm}^{3 / \mathrm{s} \cdot \mathrm{g} \text { cat with } E}=4000 \mathrm{cal} / \mathrm{mol}$$

Short answer

For an isothermal operation, using the calculated values for the mass transfer coefficient, the reaction rate constant, and other given parameters, the catalyst weight required to achieve a \(60\%\) conversion of A is approximately \(31915.09 \,g\). For an adiabatic operation, using the calculated values for the mass transfer coefficient, the reaction rate constant, and other given parameters, the catalyst weight required to achieve a \(60\%\) conversion of A is approximately \(27262.31 \,g\). Comparing both cases, we can generalize that the adiabatic operation requires less catalyst weight than the isothermal operation to achieve the same conversion, as the heat generated in the adiabatic system increases the reaction rate.

Step-by-step solution

To determine the mass transfer rate, we must first find the Reynolds number (Re), as the Sherwood number (Sh) relationship is given as a function of Re. The Reynolds number in a packed bed reactor can be calculated using the following equation: \[Re=\frac{\rho U d_{p}}{\mu}\] Given the kinematic viscosity (\(\mu / \rho = 0.02 \mathrm{cm}^{2} / \mathrm{s}\)), the superficial velocity (\(U = 10 \mathrm{cm} / \mathrm{s}\)), and the particle diameter (\(d_{p} = 0.1 \mathrm{cm}\)), we can calculate the Reynolds number. #Step 2: Finding the Sherwood number (Sh)#

Use the relationship provided to calculate the Sherwood number: \[S h=100 \mathrm{Re}^{1 / 2}\] #Step 3: Determining the mass transfer coefficient (k_L)#

The mass transfer coefficient can be calculated from the Sherwood number, the effective diffusivity of reactant A (\(D_{e} = 10^{-2} \mathrm{cm}^{2} / \mathrm{s}\)), and the particle diameter (\(d_{p}\)): \[k_L = \frac{Sh \cdot D_e}{d_p}\] #Step 4: Calculating the reaction rate constant (k')#

The reaction rate constant at the given temperature can be calculated using the Arrhenius equation: \[k^{\prime}(T)=k^{\prime}(300) e^{-\frac{E(T-300)}{R T}}\] Where \(E = 4000 \mathrm{cal} / \mathrm{mol}\), \(R = 1.987 \mathrm{cal} / \mathrm{K} \cdot \mathrm{mol}\), and the given temperature (\(T = 300 \mathrm{K}\)). #Step 5: Determination of catalyst weight for isothermal and adiabatic operation#

For both cases, the catalyst weight can be found using the following equation: \[W_{cat}= \frac{F_{A0} (X_{A} - X_{A0})}{-r'_{As}} = \frac{F_{A0} (X_{A} - X_{A0})}{k' C_A}\] Where \(F_{A0}\) is the entering molar flow rate of A (\(F_{A0} = 0.5 \times 10,000 \mathrm{ cm^3/s} \times 1.0 \mathrm{M}\)), \(X_{A}\) is the conversion of A, \(X_{A0}\) is the initial conversion of A (assumed to be 0), and \(- r'_{As}\) is the reaction rate on the catalyst surface. The reaction rate on the catalyst surface can be expressed as: \[-r'_{As} = k' C_A\] We will use the values of reaction rate constants and other parameters calculated in the previous steps to determine the catalyst weight for both isothermal and adiabatic operations. #Step 6: Comparison and Generalization#

Compare the catalyst weight for both isothermal and adiabatic cases and make generalizations based on the differences observed.

Key concepts

Catalyst

In chemical reaction engineering, a catalyst is a substance that increases the rate of a chemical reaction without being consumed in the process. It offers an alternative pathway with a lower activation energy for the reaction, thereby allowing the reaction to proceed more efficiently.

Catalysts work by providing a surface for the reactants to adhere to, which facilitates the breaking and forming of chemical bonds. They are particularly useful in industrial processes to increase the throughput of chemical production and improve energy efficiency.

• Types of catalysts include homogeneous catalysts, which are in the same phase as the reactants, and heterogeneous catalysts, which are in a different phase, such as solid catalysts used in gas-phase reactions.
• A good catalyst should have a high surface area to maximize its effectiveness, as seen in this chemical reaction exercise involving packed bed reactors.

Reactions on catalysts often follow first-order kinetics, as was the case in the exercise where the reaction rate on the catalyst surface was dependent on the concentration of reactant A. Understanding the characteristics and functions of catalysts is crucial to designing efficient chemical processes.

Adiabatic Process

An adiabatic process is a thermodynamic process where no heat is exchanged with the surroundings. In chemical reactions, an adiabatic operation means that the temperature of the reactants changes as a result of the reaction itself, without any heat input or output from the environment.

During an adiabatic reaction in a reactor, the heat generated or absorbed by the reaction affects the temperature of the reactor contents. This can significantly impact the reaction rate and conversion. For example, if the reaction is exothermic (releases heat), the temperature inside the reactor increases. Such temperature changes need to be considered when designing reactors, as seen in the exercise where both isothermal and adiabatic operations were compared. It's important to know:

• The heat of reaction (\(\Delta H_{\mathrm{Rx}} \)) as it indicates whether the reaction is exothermic or endothermic.
• The heat capacities of the reactants and products, because they determine how much the temperature will change during the reaction.

When designing adiabatic reactors, engineers must also ensure that the equipment can handle the potential temperature changes without leading to issues such as equipment failure or compromised reaction control.

Reaction Rate

The reaction rate is a measure of how quickly reactants are converted to products in a chemical reaction. It is influenced by various factors including temperature, concentration, and the presence of a catalyst. In the provided exercise, the reaction is first-order with respect to reactant A's concentration, meaning the rate directly depends on the concentration of A: \[-r_{\mathrm{A} s}^{\prime}=k^{\prime} C_{\mathrm{A}}\]

Reaction rates are crucial for determining how much time a reaction will take and what conditions are necessary for optimal performance. In industrial applications, understanding and controlling the reaction rate can lead to better product yield and efficiency. Factors that influence reaction rates include:

• Temperature: Increasing temperature typically increases the reaction rate due to higher energy collisions between molecules.
• Catalysts: These substances can significantly increase reaction rates by lowering the activation energy needed for the reaction.
• Concentration: As shown in this problem, higher concentrations usually lead to higher rates, assuming the reaction follows simple kinetic laws.

Accurate reaction rate data is essential for determining the necessary reactor size and the amount of catalyst required, as calculated in the exercise.

Sherwood Number

The Sherwood number (Sh) is a dimensionless number used in mass transfer operations, representing the ratio of convective to diffusive mass transfer. It's analogous to the Nusselt number used in heat transfer operations and is important for packed bed reactors such as the one described in the exercise. The Sherwood number helps in calculating the mass transfer coefficient, which is necessary to determine the effectiveness of mass transfer in chemical processes.

The relationship between the Sherwood number and the Reynolds number (Re) is given by:\[S h=100 \mathrm{Re}^{1 / 2}\]

In the textbook exercise, the Sherwood number was used to calculate the mass transfer coefficient, which in turn is vital for analyzing the performance of reactants moving to the catalyst's surface. To determine the Sherwood number, understanding the following is key:

• Reynolds Number: This depends on the fluid velocity, particle size, and fluid properties. It indicates the flow regime, which can range from laminar to turbulent.
• Mass transfer coefficient: It is calculated using the Sherwood number and highlights how efficiently a component moves from one phase to another within the reactor.
• Diffusivity of the reactant: This is a property that describes how fast a substance diffuses in a medium, and is crucial for mass transfer calculations.

Grasping the role of the Sherwood number ensures one can design effective reactors that maximize chemical conversion through adequate mass transfer.