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 \(\mu=\sigma_{\mathrm{H}_{2} / \mathrm{CO} /(3-\alpha) \text
{. }}\)
