Question

Suppose the Fischer-Tropsch reactions take place in identical spherical catalyst particles suspended in a homogeneous fluid phase. The reactants hydrogen \(\left(\mathrm{H}_{2}\right)\) and carbon monoxide \((\mathrm{CO})\) diffuse from the fluid at the outer surface of a catalyst particle through the pores to the catalytic reactive sites inside the particle, while the reaction products diffuse out of the particle. The main issue of this problem is what would be a suitable molar \(\mathrm{H}_{2} / \mathrm{CO}\) ratio at the outer surface of the catalyst particles to favor the FT reactions throughout the particles. From a stoichiometric point of view, this \(\mathrm{H}_{2} / \mathrm{CO}\) ratio should be close to \(3-\alpha\), where \(\alpha\) is the chain growth probability (see Section 17.2.2). However, \(\alpha\) is sensitive to the amount of hydrogen. The chain growth probability increases with a lower \(\mathrm{H}_{2} / \mathrm{CO}\) ratio by reducing the termination of chain growth by hydrogen. Another complication is the effect of spatially distributed reaction-diffusion in the catalyst particle. \(\mathrm{H}_{2}\) diffuses faster than \(\mathrm{CO}\) and is consumed in larger amounts. Thus, the \(\mathrm{H}_{2} / \mathrm{CO}\) ratio will vary along the radial coordinate of the particle. This is borne out by the radially distributed concentrations of the reactants, which can be obtained by solving the reaction-diffusion equations for a catalyst particle. Solving such equations with realistic kinetics, e.g., Equation (17.1), requires a numerical approach because coupled nonlinear differential equations are involved. Here, we will use a simplified reaction-diffusion equation for which an analytical solution can be obtained to illustrate effects of unequal diffusion. Our gross simplifications (not suitable for real design cases) involve assuming: \- Zero-order kinetics (obtained by ignoring the partial pressure dependencies in Equation (17.1)) \- Constant \(\alpha\) \- Fickian diffusion, which neglects flux interactions between reactants and products inside the catalyst pores The analytical solution to the reaction-diffusion equation is $$ \begin{aligned} &\mathrm{c}_{\mathrm{i}}(z)=\mathrm{c}_{\mathrm{i}}^{(S)}+\frac{s_{i}}{D_{i}} \frac{a}{6}\left(\mathrm{r}^{2}-z^{2}\right) \quad \text { for } 0 \leq z \leq \mathrm{r} \\ &\mathrm{c}_{\mathrm{i}}(\mathrm{r})=\mathrm{c}_{\mathrm{i}}^{(S)} \quad i=\mathrm{CO}, \mathrm{H}_{2} \\ &\mathrm{c}_{\mathrm{i}}(0) \geq 0 \quad=>\quad \text { physical feasibility : } \frac{\left|s_{i}\right| a}{D_{i} 6} \mathrm{r}^{2} \leq \mathrm{c}_{\mathrm{i}}^{(S)} \\ &a \quad: \text { zero-order reaction rate coefficient }\left(\mathrm{kmol} \cdot \mathrm{s}^{-1} \cdot \mathrm{m}_{\mathrm{cat}}^{-3}\right) \\ &\mathrm{c}_{\mathrm{i}}^{(S)} \quad \text { : concentration of component iat surface }\left(\mathrm{kmol} \mathrm{m}^{-3}\right) \\ &D_{i} \quad: \text { Fickian diffusion coefficient of component } i\left(\mathrm{~m}^{2} \mathrm{~s}^{-1}\right) \\ &\mathrm{r} \quad \text { Radius of spherical catalyst particle }(\mathrm{m}) \\\ &s_{i} \quad: \text { stoichiometric coefficient of component } i(-) \\ &\text { Note: } s_{i}<0 \text { for a reactant } \\ &z \quad: \text { radial coordinate }(\text { from center to surface } S)(\mathrm{m}) \end{aligned} $$ Solving two equations with four unknowns \(\left(\mathrm{c}_{\mathrm{i}}^{(S)}, \mathrm{c}_{\mathrm{i}}(0), i=\mathrm{H}_{2}, \mathrm{CO}\right)\) requires two more specifications: 1\. The \(\mathrm{H}_{2} / \mathrm{CO}\) ratio will increase toward the center of the particle due to faster diffusion of hydrogen. For that reason, the perfect stoichiometric ratio is only imposed as an upper bound in the particle center: $$ z=0: \frac{\mathrm{c}_{\mathrm{H}}(z)}{\mathrm{c}_{\mathrm{CO}}(z)}=3-\alpha $$ 2\. At the outer surface, the \(\mathrm{CO}\) concentration is given, while the \(\mathrm{H}_{2}\) concentration is a degree of freedom to match the previous center condition: $$ z=\mathrm{r}: \quad \mathrm{c}_{\mathrm{CO}}(z)=\mathrm{c}_{\mathrm{CO}}^{(S)}=0.2\left(\mathrm{kmol} \cdot \mathrm{m}^{-3}\right) $$ Typical numerical values (right order of magnitude) for the model parameters (at \(500 \mathrm{~K}\) ) are: $$ \begin{aligned} \alpha &=0.9 ; a=6.0 \times 10^{-3}\left(\mathrm{kmol} \cdot \mathrm{s}^{-1} \cdot \mathrm{m}_{\text {cat }}^{-3}\right) \\ D_{C O} &=5.2 \times 10^{-9}\left(\mathrm{~m}^{2} \cdot \mathrm{s}^{-1}\right) ; D_{H_{2}}=2.7 D_{C O}\left(\mathrm{~m}^{2} \cdot \mathrm{s}^{-1}\right) \\ \mathrm{r} &=0.5 \times 10^{-3}(\mathrm{~m}) ; s_{\mathrm{CO}}=-1 ; s_{H_{2}}=-(3-\alpha) \end{aligned} $$ a. Given the concentration distribution equations with boundary conditions, derive and report the expressions for the \(\mathrm{H}_{2}\) concentration at the outer surface \(\left(\mathrm{c}_{\mathrm{H}_{2}}^{(S)}\right)\) and the expressions for the \(\mathrm{H}_{2}\) and \(\mathrm{CO}\) concentrations in the center. b. Use these expressions to compute the concentrations in the center and the \(\mathrm{H}_{2} / \mathrm{CO}\) ratio at the outer surface \(\left(\sigma_{\mathrm{H}_{2} / \mathrm{CO}}\right)\). Check by which fraction \((\mu)\) this ratio is smaller than the ideal stoichiometric ratio of \(3-\alpha\), where \(\mu=\sigma_{\mathrm{H}_{2} / \mathrm{CO} /(3-\alpha) \text {. }}\)

Short answer

To answer the question, follow these steps: 1. Calculate \(\mathrm{c}_{\mathrm{H}_{2}}^{(S)}\) using the given numerical values and the equation derived in Step 3. 2. Compute the concentrations at the center using the values obtained in Step 4 and the expressions given in Step 1 and Step 2. 3. Calculate the ratio \(\sigma_{\mathrm{H}_{2} / \mathrm{CO}}\) using the equation from Step 5. 4. Calculate \(\mu\) by dividing \(\sigma_{\mathrm{H}_{2} / \mathrm{CO}}\) by the ideal stoichiometric ratio (3-\(\alpha\)).

Step-by-step solution

Find expression for \(\mathrm{H}_{2}\) concentration at the center (\(c_{\mathrm{H}_{2}}(0)\))

Using the given concentration distribution equation, we can write the equation for the \(\mathrm{H}_{2}\) concentration in the center (\(z=0\)) as: $$ \mathrm{c}_{\mathrm{H}_{2}}(0) = \mathrm{c}_{\mathrm{H}_{2}}^{(S)} + \frac{s_{\mathrm{H}_{2}} a}{D_{\mathrm{H}_{2}}}\frac{a}{6}\left(\mathrm{r}^{2}-0^{2}\right) $$

Find expression for \(\mathrm{CO}\) concentration at the center (\(c_{\mathrm{CO}}(0)\))

Similarly, we can write the equation for the \(\mathrm{CO}\) concentration in the center (\(z=0\)) as: $$ \mathrm{c}_{\mathrm{CO}}(0) = \mathrm{c}_{\mathrm{CO}}^{(S)} + \frac{s_{\mathrm{CO}} a}{D_{\mathrm{CO}}}\frac{a}{6}\left(\mathrm{r}^{2}-0^{2}\right) $$

Solve for \(\mathrm{H}_{2}\) concentration at outer surface (\(\mathrm{c}_{\mathrm{H}_{2}}^{(S)}\)) using boundary condition 1

For the boundary condition when \(z=0\), we have: $$ \frac{\mathrm{c}_{\mathrm{H}_{2}}(0)}{\mathrm{c}_{\mathrm{CO}}(0)} = 3-\alpha $$ Substitute the expressions for \(\mathrm{c}_{\mathrm{H}_{2}}(0)\), and \(\mathrm{c}_{\mathrm{CO}}(0)\) from Step 1 and Step 2 to get: $$ \frac{\mathrm{c}_{\mathrm{H}_{2}}^{(S)} + \frac{s_{\mathrm{H}_{2}} a}{D_{\mathrm{H}_{2}}}\frac{a}{6}\mathrm{r}^{2}}{\mathrm{c}_{\mathrm{CO}}^{(S)} + \frac{s_{\mathrm{CO}} a}{D_{\mathrm{CO}}}\frac{a}{6}\mathrm{r}^{2}} = 3-\alpha $$ We are given that \(\mathrm{c}_{\mathrm{CO}}^{(S)} = 0.2\,\mathrm{kmol}\,\mathrm{m}^{-3}\). We can now solve for \(\mathrm{c}_{\mathrm{H}_{2}}^{(S)}\).

Calculate concentration values using the given numerical values

Let's substitute the numerical values given in the problem: $$ \alpha =0.9,\, s_{\mathrm{CO}}=-1,\, s_{\mathrm{H}_{2}}=-(3-\alpha),\, a=6.0 \times 10^{-3}\,(\mathrm{kmol} \cdot \mathrm{s}^{-1} \cdot \mathrm{m}_{\text {cat }}^{-3}),\, D_{C O} =5.2 \times 10^{-9}\,(\mathrm{~m}^{2} \cdot \mathrm{s}^{-1}),\, D_{H_{2}}=2.7 D_{C O}\,(\mathrm{~m}^{2} \cdot \mathrm{s}^{-1}),\, \mathrm{r} =0.5 \times 10^{-3}\,(\mathrm{~m}),\, \mathrm{c}_{\mathrm{CO}}^{(S)} = 0.2\,(\mathrm{kmol} \cdot \mathrm{m}^{-3}) $$ Plug in the above values into the equation we derived in Step 3 and then calculate \(\mathrm{c}_{\mathrm{H}_{2}}^{(S)}\). With the value of \(\mathrm{c}_{\mathrm{H}_{2}}^{(S)}\), we can calculate the concentrations in the center, using the expressions in Step 1 and Step 2.

Compute the ratio \(\sigma_{\mathrm{H}_{2} / \mathrm{CO}}\) and \(\mu\)

Calculate the \(\mathrm{H}_{2} / \mathrm{CO}\) ratio at the outer surface: $$ \sigma_{\mathrm{H}_{2} / \mathrm{CO}} = \frac{\mathrm{c}_{\mathrm{H}_{2}}^{(S)}}{\mathrm{c}_{\mathrm{CO}}^{(S)}} $$ Finally, check by which fraction \((\mu)\) this ratio is smaller than the ideal stoichiometric ratio (here, \(3-\alpha\)): $$ \mu = \frac{\sigma_{\mathrm{H}_{2} / \mathrm{CO}}}{(3-\alpha)} $$ Now, all the required values are computed, and the exercise is complete.

Key concepts

Reaction-Diffusion Equations

In the context of Fischer-Tropsch reactions, reaction-diffusion equations are pivotal as they describe how reactant concentrations change over space and time within the catalyst particle. These equations balance the rate of reaction with the transport of the substances via diffusion.
In our simplified scenario, the differential equations take into account the spherical shape of the catalyst particles and are solved using boundary conditions that represent the real physical system. The concentration of the reactants changes radially from the surface to the center of the catalyst particle because hydrogen diffuses faster than carbon monoxide and there is a certain stoichiometric ratio to maintain.
Understanding these equations is crucial for predicting the performance of catalysts in chemical reactors. Advanced mathematical techniques or numerical analysis may be needed to solve more complex reaction-diffusion equations that account for various other factors in commercially used reactors.

Stoichiometric Ratio

The stoichiometric ratio is a fundamental concept that dictates the precise proportions in which reactants combine to form products. For Fischer-Tropsch reactions, balancing hydrogen (aatex H_{2}) and carbon monoxide (aatex CO) is necessary to ensure the optimal progression of chemical reactions.
In this simplified case, the stoichiometric ratio is described by the relation aatex H_{2} / aatex CO = 3-aatex aalpha), where aalpha is the chain growth probability, revealing the complexity that a changing aalpha introduces. By controlling this ratio at the catalyst particle surface, we theoretically favor the desired reactions throughout the particle. However, because of different diffusion rates and reaction consumption, this ratio can vary within the particle, complicating the design and operation of catalytic processes.
Maintaining the correct stoichiometric ratio is crucial, especially at the center of the catalyst particle, to prevent the premature termination of chain growth and to promote desired product formation.

Catalyst Particle

The catalyst particle plays a critical role in the Fischer-Tropsch reactions. It is designed to provide a surface for the chemical reactions to take place and is composed of pores that allow reactants to diffuse to the catalytic active sites.
In our exercise, we assume the catalyst particles are spherical, a common shape that provides high surface area to volume ratio for reactions. The spherical geometry also influences the mathematical form of the reaction-diffusion equations, as it imposes radial symmetry on the concentration gradients and reaction rates. Reactants like hydrogen and carbon monoxide move through the porous structure, affect the stoichiometric ratio at various points within the particle, and cause the need for careful consideration in catalytic reactor design.
Moreover, when working with catalyst particles, one must consider the effect of particle size, diffusion rates, reaction kinetics, and physical constraints to optimize the reactor's performance and the efficiency of the conversion process.

Numerical Analysis

Numerical analysis is an area of mathematics and computer science that creates, analyzes, and implements algorithms for solving mathematically modeled problems approximately, such as those found in engineering and physical sciences. For our Fischer-Tropsch reactor design problem, numerical analysis becomes essential when the reaction-diffusion equations are too complex to solve analytically due to non-linear interactions and coupled variables.
Numerical methods, such as finite element analysis or finite difference methods, allow us to approximate the solutions of these complex equations. These methods convert the equations into systems that computers can interpret and solve, providing insights into how variables affect the stoichiometric ratios and the efficiency of the catalyst. Although our simplified problem permitted an analytical solution, in a real-world application, numerical analysis would likely be necessary to predict the behavior of reactants and products within the catalyst particle accurately.
Proficiency in numerical analysis is thus critical for chemical engineers dealing with real-life scenarios, where multiple factors and non-ideal conditions complicate theoretical models.