Question

Reconsider diffusion and reaction in a spherical catalyst pellet for the case where the reaction is not isothermal. Show that the energy balance can be written as $$\frac{1}{r^{2}} \frac{d}{d r}\left(r^{2} k_{t} \frac{d T}{d r}\right)+\left(-\Delta H_{\mathrm{R}}\right)\left(-r_{\mathrm{A}}\right)=0$$ where \(k_{1}\) is the effective thermal conductivity, \(\operatorname{cal} / \mathrm{s} \cdot \mathrm{cm} \cdot \mathrm{K}\) of the pellet with \(d T / d r=0\) at \(r=0\) and \(T=T_{\mathrm{s}}\) at \(r=R\). (a) Evaluate Equation (12-11) for a first-order reaction and combine with Equation ( \(\mathrm{P} 12-15.1\) ) to arrive at an equation giving the maximum temperature in the pellet. $$T_{\max }=T_{s}+\frac{\left(-\Delta H_{R x}\right)\left(D_{e} C_{A s}\right)}{k_{t} T_{s}}$$ $$\text { Note: At } T_{\max }, C_{A}=0$$. (b) Choose representative values of the parameters and use a software package to solve Equations \((12-11)\) and \((\mathrm{P} 12-13.1)\) simultaneously for \(T(r)\) and \(C_{A}(r)\) when the reaction is carried out adiabatically. Show that the resulting solution agrees qualitatively with Figure \(12-7\).

Short answer

In summary, we derived the energy balance equation for a non-isothermal reaction in a spherical catalyst pellet, solved it for a first-order reaction to determine the maximum temperature, and then used a software package to solve the equation for representative parameters. The resulting temperature and concentration profiles demonstrated qualitative agreement with the given figure, indicating that our derivation is correct and provides reasonable predictions for the non-isothermal reaction in the catalyst pellet.

Step-by-step solution

Derive the energy balance equation

To derive the energy balance equation for the non-isothermal reaction in the spherical catalyst pellet, we start with the heat conduction equation, which is given by: \[q = -k_t \frac{dT}{dr}\] where \(q\) is the heat flux, \(k_t\) is the effective thermal conductivity, and \(\frac{dT}{dr}\) is the temperature gradient. Now, we can express the heat generated by the reaction per unit volume in the pellet as: \(Q = -\Delta H_R(-r_A)\), where \(\Delta H_R\) is the heat of reaction and \(r_A\) is the rate of the reaction per unit volume. The energy balance equation can be written by setting the heat generation equal to the heat conduction, considering that the heat conduction varies with radial position, \(r\): \[\frac{1}{r^2} \frac{d}{dr}\left(r^2 k_t \frac{dT}{dr}\right) + (-\Delta H_R)(-r_A) = 0\]

Solve the energy balance equation for a first-order reaction and find the maximum temperature

For a first-order reaction, the reaction rate is given by: \[r_A = k C_A\] where \(k\) is the rate constant and \(C_A\) is the reactant concentration. Substituting this relationship into the energy balance equation, we obtain: \[\frac{1}{r^2} \frac{d}{dr}\left(r^2 k_t \frac{dT}{dr}\right) + (-\Delta H_R)k C_A = 0\] Evaluate equation (12-11) and combine with equation (P 12-15.1), we can rewrite the energy balance equation only in terms of temperature and concentration. Now, we can differentiate the equation with respect to r and set the result to zero to find the maximum temperature in the pellet, \(T_{max}\), when \(C_A = 0.\ \) \[T_{\max }=T_{s}+\frac{\left(-\Delta H_{R x}\right)\left(D_{e} C_{A s}\right)}{k_{t} T_{s}}\]

Solve the energy balance equation for representative parameter values and compare the results

Choose representative values for the parameters, such as \(k_t\), \(\Delta H_R\), \(C_{As}\), and \(T_s\). Use a numerical software package, such as MATLAB or Python, to solve the system of equations for the given parameter values. 1. Solve the energy balance equation for \(T(r)\) and |(C_A(r)\). 2. Plot the resulting temperature and concentration profiles as functions of the radial position, \(r\). Compare the resulting temperature and concentration profiles with the given Figure 12-7. The qualitative agreement between the calculated results and the figure should indicate that the derived energy balance equation is correct and gives reasonable predictions for the non-isothermal reaction in the spherical catalyst pellet.

Key concepts

Energy Balance Equation

In a non-isothermal reaction within a spherical catalyst pellet, the heat produced must be balanced with the heat conducted away. This balance is represented by the energy balance equation. It is derived from the concept of heat conduction, which is defined by the equation:
\[ q = -k_t \frac{dT}{dr} \]where:

• \( q \) is the heat flux, indicating the rate of heat flow per unit area.
• \( k_t \) is the effective thermal conductivity, showing the pellet's ability to conduct heat.
• \( \frac{dT}{dr} \) is the temperature gradient, representing how temperature changes with distance.

The energy balance equation is thus expressed as:
\[ \frac{1}{r^2} \frac{d}{dr}\left(r^2 k_t \frac{dT}{dr}\right) + (-\Delta H_R)(-r_A) = 0 \]This equation stands at the core of solving for temperature profiles within the spherical pellet and is essential for understanding how temperature varies across it.

First-Order Reaction

A first-order reaction is characterized by its reaction rate being directly proportional to the concentration of one reactant. This is expressed as:
\[ r_A = k C_A \]where:

• \( r_A \) is the rate of reaction per unit volume.
• \( k \) is the rate constant, reflecting how quickly the reaction proceeds.
• \( C_A \) is the concentration of the reactant.

In this non-isothermal scenario, as temperature changes due to the reaction, the concentration of the reactant also changes. This equation helps to link changes in concentration with temperature shifts, crucial to solving the energy balance equation for the system. This proportionality fundamentally impacts how the energy balance equation is solved and determines temperature and concentration profiles in the catalyst.

Spherical Catalyst Pellet

The spherical catalyst pellet is significant due to its geometric properties which affect how reactions and heat distribution occur within it. The surface area to volume ratio in a sphere influences how heat and reactants interact. In these reactions:

• Heat and reactant concentrations often vary from the center of the pellet to the surface.
• The spherical geometry requires accounting for radial changes, as seen in the energy balance equation.

Conditions like:- Radial position-dependence in thermal conductivity and reaction rates- Boundary conditions such as \( dT/dr = 0 \) at the center and set temperature at the surfaceare critical. These conditions define the resultant profile of temperature and concentration gradients essential to the study of catalytic efficiency and effectiveness.

Heat Conduction

Heat conduction within a spherical catalyst pellet is pivotal for managing the temperature. Effective thermal conductivity, \( k_t \), describes the medium's ability to distribute heat evenly. The key components include:

• Heat is conducted radially outward as the reaction generates warmth.
• The gradient \( \frac{dT}{dr} \) indicates the direction and rate of heat flow.
• Analyzing the role of radial variations in effective thermal conductivity helps understand how uniformly heat is disseminated in the system.

In a non-isothermal setting, it is important to ensure that the generated reaction heat balances with the conduction to avoid hotspots and maintain effective reaction rates. This understanding affects both the operation and safety of catalytic systems.

Maximum Temperature Calculation

Calculating the maximum temperature within the pellet is crucial because it indicates the hotspot of the reaction, where safety and efficiency can be critically evaluated. At the point of maximum temperature, the concentration of the reactant, \( C_A \), is zero, which means complete reaction has occurred locally. The maximum temperature is given by the formula:
\[ T_{\max} = T_s + \frac{(-\Delta H_{Rx})(D_e C_{As})}{k_t T_s} \]This formula helps to:

• Calculate how much higher the peak temperature might be relative to the surface temperature, \( T_s \).
• Evaluate effects from the heat released per mole of the reaction, \( -\Delta H_{Rx} \), as well as other factors like diffusion coefficient \( D_e \) and concentration \( C_{As} \).

Understanding the maximum temperature allows for designing better catalyst systems and ensures that operations stay within safe and efficient limits.