7.3 Diffusion with Chemical Reaction

Diffusion with chemical reaction is a crucial concept in heat and mass transfer. It explores how substances move and react simultaneously, affecting concentration gradients and overall system behavior. This topic bridges the gap between simple diffusion and complex real-world processes.

Understanding diffusion-reaction coupling is key to many applications, from catalysis to biological systems. We'll examine how reaction rates and diffusion rates interact, and learn to solve for concentration profiles and effectiveness factors in steady-state systems.

Diffusion and Reaction Coupling in Steady-State Systems

Interaction between Diffusion and Chemical Reactions

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• Diffusion and chemical reactions are often coupled in real-world systems, where the rate of diffusion influences the rate of reaction and vice versa
• The coupling of diffusion and reaction can lead to the formation of concentration gradients, which drive the diffusive transport of species (oxygen and glucose in biological tissues)
• In steady-state systems, the concentration profiles of reactants and products remain constant over time, but vary spatially due to the combined effects of diffusion and reaction (catalytic converters in automobiles)

Factors Determining System Behavior

• The relative rates of diffusion and reaction determine the overall behavior of the system, such as whether the process is diffusion-limited or reaction-limited
  • Diffusion-limited systems have slow diffusion rates compared to reaction rates, resulting in concentration gradients (gas absorption in liquid films)
  • Reaction-limited systems have fast diffusion rates compared to reaction rates, resulting in uniform concentrations (enzymatic reactions in well-mixed solutions)
• The effectiveness factor, η, is a measure of the extent to which diffusion limitations affect the overall reaction rate in a porous catalyst pellet
  • η ranges from 0 to 1, with values closer to 1 indicating minimal diffusion limitations (small catalyst pellets with high porosity)
  • η values closer to 0 indicate significant diffusion limitations (large catalyst pellets with low porosity)

Deriving the Reaction-Diffusion Equation

General Form of the Reaction-Diffusion Equation

• The reaction-diffusion equation describes the spatial and temporal variation of species concentrations in a system where both diffusion and chemical reactions occur
• The general form of the reaction-diffusion equation is: $\frac{\partial C}{\partial t} = D \nabla^2 C + R(C)$

where $C$ is the concentration, $t$ is time, $D$ is the diffusion coefficient, $\nabla^2$ is the Laplacian operator, and $R(C)$ represents the reaction term

Steady-State Simplification

• For a steady-state system with one-dimensional diffusion and a first-order reaction, the reaction-diffusion equation simplifies to: $D \frac{d^2C}{dx^2} = k C$

where $D$ is the diffusion coefficient, $C$ is the concentration, $x$ is the spatial coordinate, and $k$ is the first-order reaction rate constant

• The steady-state assumption eliminates the time-dependent term, $\frac{\partial C}{\partial t}$, from the general reaction-diffusion equation
• The one-dimensional simplification reduces the Laplacian operator to a second-order derivative with respect to $x$

Solving the Reaction-Diffusion Equation

• The reaction-diffusion equation can be solved analytically or numerically, depending on the boundary conditions and the complexity of the reaction kinetics
  • Analytical solutions are possible for simple geometries and boundary conditions (planar, cylindrical, or spherical systems with constant surface concentrations)
  • Numerical methods (finite difference, finite element) are required for complex geometries or non-linear reaction kinetics
• The solution to the reaction-diffusion equation provides the concentration profile of the species as a function of position in the system
• The reaction-diffusion equation can be extended to more complex systems, such as those with multiple species, higher-order reactions, or non-linear kinetics (Michaelis-Menten kinetics for enzyme-catalyzed reactions)

Solving for Concentration Profiles and Rates

Concentration Profiles

• The concentration profile in a steady-state diffusion-reaction system can be obtained by solving the reaction-diffusion equation with appropriate boundary conditions
• For a first-order reaction in a planar system with constant surface concentrations, the concentration profile is given by: $C(x) = C_0 \frac{\cosh(x \sqrt{k/D})}{\cosh(L \sqrt{k/D})}$

where $C_0$ is the surface concentration, $L$ is the half-thickness of the planar system, and other variables are as defined previously

• The concentration profile depends on the relative magnitudes of the diffusion coefficient, $D$, and the reaction rate constant, $k$, as well as the system geometry and boundary conditions

Reaction Rates

• The reaction rate can be calculated from the concentration profile using the reaction rate law, e.g., for a first-order reaction: $r = k C(x)$

• The local reaction rate varies with position due to the concentration gradient established by the coupled diffusion and reaction processes

• The average reaction rate in the system can be determined by integrating the local reaction rate over the entire domain and dividing by the volume: $\bar{r} = \frac{1}{V} \int_V k C(x) dV$

Effectiveness Factor

• The effectiveness factor, η, can be calculated as the ratio of the actual reaction rate in the presence of diffusion limitations to the reaction rate that would occur if the entire system were exposed to the surface concentration: $\eta = \frac{\bar{r}}{k C_0}$

• η ranges from 0 to 1, with values closer to 1 indicating minimal diffusion limitations and values closer to 0 indicating significant diffusion limitations

• The effectiveness factor is a useful metric for assessing the impact of diffusion on the overall reaction rate and for optimizing catalyst design (pellet size and porosity)

Reaction Kinetics vs Diffusion Limitations

Thiele Modulus

• The relative importance of reaction kinetics and diffusion in determining the overall process behavior can be assessed using the Thiele modulus, φ, which compares the characteristic time scales of reaction and diffusion
• For a first-order reaction in a planar system, the Thiele modulus is given by: $\phi = L \sqrt{\frac{k}{D}}$

where $L$ is the half-thickness of the planar system, $k$ is the first-order reaction rate constant, and $D$ is the diffusion coefficient

• The Thiele modulus is a dimensionless number that relates the rate of reaction to the rate of diffusion

Reaction-Limited vs Diffusion-Limited Systems

• When φ << 1, the system is reaction-limited, meaning that the overall process is controlled by the intrinsic reaction kinetics, and concentration gradients are negligible (homogeneous catalysis)
• When φ >> 1, the system is diffusion-limited, indicating that the overall process is controlled by the rate of diffusion, and significant concentration gradients exist within the system (heterogeneous catalysis with porous catalysts)
• The effectiveness factor, η, decreases as the Thiele modulus increases, demonstrating the increasing importance of diffusion limitations on the overall reaction rate
  • For reaction-limited systems (φ << 1), η approaches 1, indicating that the actual reaction rate is close to the intrinsic reaction rate
  • For diffusion-limited systems (φ >> 1), η approaches 0, indicating that the actual reaction rate is significantly lower than the intrinsic reaction rate due to diffusion limitations

Optimizing Catalyst Design

• In porous catalyst pellets, the effectiveness factor can be used to determine the optimal pellet size and porosity that maximize the overall reaction rate while minimizing the effects of diffusion limitations
  • Smaller pellet sizes reduce the diffusion path length and increase the effectiveness factor, but may lead to higher pressure drops and manufacturing costs
  • Higher pellet porosities increase the effective diffusion coefficient and the effectiveness factor, but may reduce the mechanical strength and catalyst loading
• Optimizing catalyst design involves finding a balance between reaction kinetics and diffusion limitations to achieve the desired performance and cost targets (eggshell catalysts with active material concentrated near the surface)