7.3 Diffusion with Chemical Reaction

Diffusion with Chemical Reaction

In many real systems, species don't just diffuse — they also react as they move. Diffusion with chemical reaction describes how these two processes interact simultaneously, shaping concentration gradients and controlling overall system performance. This coupling shows up everywhere from catalytic reactors to oxygen transport in biological tissue.

The central challenge is figuring out whether diffusion or reaction is the bottleneck. Solving that question requires deriving concentration profiles, calculating effectiveness factors, and using the Thiele modulus to characterize system behavior.

Diffusion and Reaction Coupling in Steady-State Systems

Interaction between Diffusion and Chemical Reactions

Diffusion and chemical reactions are tightly coupled in most real systems. A reaction consumes (or produces) a species at a particular location, which creates a concentration gradient. That gradient then drives diffusive transport toward or away from the reaction site.

Consider oxygen diffusing into biological tissue: cells consume oxygen, lowering the local concentration and pulling more oxygen inward by diffusion. In a catalytic converter, reactant gases diffuse into the catalyst pores while simultaneously reacting on the catalyst surface.

In steady-state systems, concentration profiles don't change with time, but they do vary with position. The spatial variation reflects the balance between how fast species diffuse in and how fast they're consumed by reaction.

Factors Determining System Behavior

The relative rates of diffusion and reaction determine whether the system is diffusion-limited or reaction-limited:

• Diffusion-limited: The reaction is fast relative to diffusion, so concentration drops significantly before species can penetrate deep into the system. Steep concentration gradients form. Example: gas absorption into a liquid film where the dissolved gas reacts quickly.
• Reaction-limited: Diffusion is fast relative to reaction, so species distribute nearly uniformly before reacting. Concentration gradients are negligible. Example: a slow enzymatic reaction in a well-mixed solution.

The effectiveness factor $\eta$ quantifies how much diffusion limitations reduce the overall reaction rate inside a porous catalyst pellet:

• $\eta$ ranges from 0 to 1
• $\eta \approx 1$: diffusion is fast enough that the entire pellet interior "sees" nearly the surface concentration (small, highly porous pellets)
• $\eta \approx 0$: diffusion is so slow that most of the pellet interior is starved of reactant (large, low-porosity pellets)

Deriving the Reaction-Diffusion Equation

General Form of the Reaction-Diffusion Equation

The reaction-diffusion equation accounts for both diffusive transport and local reaction in a single expression:

$\frac{\partial C}{\partial t} = D \nabla^2 C + R(C)$

where $C$ is the species concentration, $t$ is time, $D$ is the diffusion coefficient, $\nabla^2$ is the Laplacian operator, and $R(C)$ is the volumetric reaction rate (negative for consumption, positive for production).

Steady-State Simplification

At steady state, concentrations don't change with time, so $\frac{\partial C}{\partial t} = 0$. For one-dimensional diffusion with a first-order irreversible reaction (where the species is consumed), the equation becomes:

$D \frac{d^2C}{dx^2} = kC$

Here $k$ is the first-order reaction rate constant and $x$ is the spatial coordinate. Notice the sign convention: the reaction term appears on the right because the reaction consumes species, so $R(C) = -kC$, and rearranging gives the form above.

To arrive at this simplified form:

1. Start with the general reaction-diffusion equation.
2. Set $\frac{\partial C}{\partial t} = 0$ (steady state).
3. Replace $\nabla^2 C$ with $\frac{d^2C}{dx^2}$ (one-dimensional geometry).
4. Substitute $R(C) = -kC$ for a first-order consumption reaction.

Solving the Reaction-Diffusion Equation

This is a second-order linear ODE, and its solution depends on the geometry and boundary conditions:

• Analytical solutions exist for simple geometries (planar slabs, long cylinders, spheres) with straightforward boundary conditions like constant surface concentration or symmetry at the center.
• Numerical methods (finite difference, finite element) are needed for complex geometries, non-linear kinetics, or coupled multispecies systems.

The equation extends naturally to more complex scenarios. For example, Michaelis-Menten kinetics replaces the linear $kC$ term with $\frac{V_{max} C}{K_m + C}$, which makes the equation non-linear and typically requires numerical solution.

Solving for Concentration Profiles and Rates

Concentration Profiles

For a first-order reaction in a planar slab of half-thickness $L$, with both surfaces held at concentration $C_0$ and symmetry at the center ($x = 0$), the steady-state concentration profile is:

$C(x) = C_0 \frac{\cosh\left(x \sqrt{k/D}\right)}{\cosh\left(L \sqrt{k/D}\right)}$

The shape of this profile tells you a lot:

• If $k/D$ is small (slow reaction or fast diffusion), the $\cosh$ ratio stays close to 1 everywhere, and concentration is nearly uniform across the slab.
• If $k/D$ is large (fast reaction or slow diffusion), concentration drops sharply toward the center. The reactant gets consumed before it can diffuse very far in.

Reaction Rates

Once you have the concentration profile, the local reaction rate at any position follows directly from the rate law. For a first-order reaction:

$r(x) = kC(x)$

This rate varies with position because $C(x)$ varies with position. Near the surface where concentration is highest, the local rate is highest. Deep inside the slab, the rate drops as the reactant is depleted.

The average reaction rate over the entire volume is:

$\bar{r} = \frac{1}{V} \int_V kC(x) \, dV$

This integral gives you the actual overall rate of reaction occurring in the system, accounting for the non-uniform concentration.

Effectiveness Factor

The effectiveness factor compares the actual average rate to the rate you'd get if diffusion were infinitely fast (i.e., if the entire volume were at the surface concentration $C_0$):

$\eta = \frac{\bar{r}}{kC_0}$

This is a single number that captures how much performance you're losing to diffusion resistance. An $\eta$ of 0.7 means the catalyst pellet is operating at 70% of its potential because the interior doesn't have enough reactant.

The effectiveness factor is central to catalyst design: it tells you whether making pellets smaller, increasing porosity, or changing geometry would meaningfully improve performance.

Reaction Kinetics vs Diffusion Limitations

Thiele Modulus

The Thiele modulus $\phi$ is the key dimensionless number for diffusion-reaction problems. It compares the characteristic rate of reaction to the characteristic rate of diffusion. For a first-order reaction in a planar slab:

$\phi = L \sqrt{\frac{k}{D}}$

You can think of it this way: $\phi^2 = \frac{kL^2}{D}$. The numerator $kL^2$ scales with how fast reaction depletes species over the length $L$, and the denominator $D$ scales with how fast diffusion replenishes it.

• Large $\phi$: reaction is fast relative to diffusion (diffusion-limited)
• Small $\phi$: diffusion is fast relative to reaction (reaction-limited)

Notice that $\phi$ appears naturally in the concentration profile solution, since $L\sqrt{k/D} = \phi$.

Reaction-Limited vs Diffusion-Limited Systems

$\eta = \frac{\tanh(\phi)}{\phi}$

At small $\phi$, $\tanh(\phi) \approx \phi$, so $\eta \to 1$. At large $\phi$, $\tanh(\phi) \to 1$, so $\eta \to 1/\phi$, which decreases as $\phi$ grows.

Optimizing Catalyst Design

In practice, catalyst pellet design involves balancing diffusion limitations against engineering constraints:

• Smaller pellets shorten the diffusion path (reduce $L$, reduce $\phi$, increase $\eta$), but they also increase pressure drop across packed beds and can be harder to manufacture.
• Higher porosity increases the effective diffusion coefficient inside the pellet (increase $D_{eff}$, reduce $\phi$, increase $\eta$), but weakens the pellet mechanically and reduces the amount of active material per unit volume.

One clever solution is the eggshell catalyst, where active material is deposited only near the pellet surface. Since reactant concentration is highest near the surface anyway (especially at high $\phi$), this puts the catalyst where it's most useful while keeping the pellet large enough to avoid pressure drop problems.