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Directions

For reaction in a porous catalyst pellet, the reactant concentration in the pellet can be lower than the reactant concentration at the external surface because of diffusion; this lowers the reaction rate. However, the temperature within the catalyst pellet can be higher than the temperature at the external surface; this increases the reaction rate. The effectiveness factor \(\eta\) is the ratio of the rate of reaction within the pellet volume divided by the rate if the concentration of reactant and the temperature in the pellet were the same as the concentration and temperature at the external surface of the pellet.

The effectiveness factor \(\eta\) is plotted versus the Thiele modulus \(\phi\), which can be changed with the slider or by clicking or dragging in the Effectiveness factor plot. The square of the Thiele modulus \(\phi^2\) can be viewed as the surface reaction rate over the diffusion rate.

The Arrhenius number \(\gamma\), which can be changed with a slider, is a measure of how strongly a reaction rate depends on temperature.

The Prater number \(\beta\), which can be changed with a slider, is approximately the heat generated by reaction divided by the ability of the pellet to conduct heat.

The dashed vertical line and the dots show the current \(\phi\) and the steady-state effectiveness factor(s) there. The two lower plots show the concentration and temperature profiles inside the pellet for each steady state.

The preset buttons load typical cases. “Compare curves for several β” overlays curves at the same γ, and the isothermal curve can be hidden.

Things to try

1. Set β = 0. Confirm η ≈ 1 for \(\phi\) < 0.3 and η ≈ 3/\(\phi\) for \(\phi\) > 3. How does the concentration profile change between these limits?
2. Choose γ = 20, β = 0.2 and sweep \(\phi\). Why can η exceed 1? Compare the pellet-center temperature with the surface temperature.
3. With γ = 20, increase β from 0 to 0.5. At what β does the curve become S-shaped? What does it mean physically when three η values exist at one \(\phi\)?
4. Use the “Multiple steady states” preset. Which of the three states would you expect to be unstable, and why?
5. Make β negative (endothermic). Why is η always below the isothermal value? Could η exceed 1?
6. At fixed β, lower γ toward 0. Why does the heat effect disappear even though β is not zero?
7. At large \(\phi\) all curves become straight lines on the log–log plot. What is the slope, and how does the apparent activation energy compare with the intrinsic one?

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Model and equations

A first-order, irreversible reaction occurs in a spherical catalyst pellet of radius \(R\) at steady state. Dimensionless variables are \(c=C/C_s\), \(\theta=T/T_s\) and \(x=r/R\), where \(C_s\) and \(T_s\) are the concentration and temperature at the pellet surface (external gradients are neglected).

Mass balance:

\[ \frac{d^2c}{dx^2}+\frac{2}{x}\frac{dc}{dx}=\phi^2\,c\,\exp\!\left[\gamma\left(1-\frac{1}{\theta}\right)\right] \]

Energy balance (Prater relation):

\[ \theta = 1+\beta\,(1-c) \]

Combining the two:

\[ \frac{d^2c}{dx^2}+\frac{2}{x}\frac{dc}{dx}=\phi^2\,c\,\exp\!\left[\frac{\gamma\beta(1-c)}{1+\beta(1-c)}\right] \]

Boundary conditions:

\[ \left.\frac{dc}{dx}\right|_{x=0}=0,\qquad c(1)=1 \]

Effectiveness factor (actual rate divided by the rate at surface conditions):

\[ \eta=\frac{3}{\phi^2}\left.\frac{dc}{dx}\right|_{x=1} \]

Isothermal limit (\(\beta=0\) or \(\gamma=0\)):

\[ \eta=\frac{3}{\phi}\left(\coth\phi-\frac{1}{\phi}\right) \]

Definitions of symbols

\(\phi = R\sqrt{k_s/D_e}\)  Thiele modulus, where \(k_s\) is the rate constant at \(T_s\) and \(D_e\) the effective diffusivity.

\(\gamma = E/(R_g T_s)\)  Arrhenius number, where \(E\) is the activation energy and \(R_g\) the gas constant.

\(\beta = (-\Delta H)\,D_e C_s/(k_e T_s)\)  Prater number, where \(-\Delta H\) is the heat of reaction and \(k_e\) the effective thermal conductivity. \(\beta>0\) is exothermic and \(\beta<0\) endothermic.

\(\eta\)  effectiveness factor; \(c\)  dimensionless concentration; \(\theta\)  dimensionless temperature; \(x\)  dimensionless radial position.

Numerical method

With \(s=\phi x\) the equation no longer contains \(\phi\). Integrating outward from a chosen center concentration \(c_0\) until \(c=1\) gives \(\phi=s^*\) and \(\eta=3\,c'(s^*)/s^*\) directly. Sweeping \(c_0\) from 0 to 1 traces the entire \(\eta\)–\(\phi\) curve, including S-shaped regions with multiple steady states. The middle solution of three is unstable. For large \(\gamma\) and \(\beta\) (roughly \(\gamma\ge 30\), \(\beta\ge 0.6\)) the curve can fold more than twice, so five or more steady states exist over a narrow range of \(\phi\); they alternate stable and unstable. The equation is integrated with an adaptive Runge–Kutta method with error control.

Assumptions: constant \(D_e\) and \(k_e\), and no external heat or mass transfer resistance.

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About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.