7.1 Diffusion and Fick's law

Mass transfer is crucial in chemical engineering, involving the movement of substances from high to low concentrations. Diffusion, a key process, occurs at the molecular level due to random motion and concentration gradients.

Fick's laws are fundamental to understanding diffusion. The first law relates diffusive flux to concentration gradient, while the second law describes concentration changes over time. These principles are essential for analyzing mass transfer in various systems.

Mass transfer principles

Fundamentals of mass transfer and diffusion

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• Mass transfer involves the movement of a substance from a region of high concentration to a region of low concentration, driven by a concentration gradient
• Diffusion is a specific type of mass transfer occurring at the molecular level, where molecules move randomly due to their kinetic energy, resulting in a net movement from high to low concentration
• Diffusive mass transfer is a spontaneous process not requiring external energy input, as it is driven by the second law of thermodynamics and the tendency towards increased entropy (entropy increases as molecules spread out)
• The rate of diffusion is proportional to the concentration gradient, with steeper gradients resulting in faster diffusion rates (more molecules moving down the gradient per unit time)

Steady-state and unsteady-state diffusion

• Steady-state diffusion occurs when the concentration gradient remains constant over time, resulting in a constant diffusive flux
• Unsteady-state diffusion involves a changing concentration gradient, leading to time-dependent diffusive fluxes
• In steady-state diffusion, the diffusive flux is independent of time, simplifying calculations
• Unsteady-state diffusion problems require the use of Fick's second law and may involve partial differential equations and numerical methods

Diffusion rate calculation

Fick's first law of diffusion

• Fick's first law states that the diffusive flux is proportional to the negative of the concentration gradient, with the proportionality constant being the diffusivity or diffusion coefficient
• The mathematical expression for Fick's first law in one dimension is $J = -D(dC/dx)$, where $J$ is the diffusive flux, $D$ is the diffusivity, and $dC/dx$ is the concentration gradient
• The negative sign in Fick's first law indicates that diffusion occurs in the direction of decreasing concentration
• Fick's first law can be applied to various systems, including gases (diffusion of pollutants in air), liquids (diffusion of solutes in solutions), and solids (diffusion of dopants in semiconductors)

Fick's second law of diffusion

• Fick's second law describes the change in concentration over time due to diffusion and is expressed as $∂C/∂t = D(∂²C/∂x²)$ for one-dimensional diffusion with constant diffusivity
• Fick's second law is a partial differential equation that relates the temporal change in concentration to the spatial change in the concentration gradient
• Solving Fick's second law requires initial and boundary conditions, which define the concentration distribution at the start and the behavior at the system boundaries (fixed concentration, no-flux, or mixed conditions)
• Analytical solutions to Fick's second law exist for simple geometries and boundary conditions, while numerical methods (finite difference, finite element) are used for more complex cases

Diffusion influencing factors

Concentration gradient and diffusivity

• The concentration gradient is the primary driving force for diffusion, with larger gradients resulting in faster diffusion rates (more molecules moving down the gradient per unit time)
• Diffusivity is a measure of the ease with which a substance diffuses through a medium and depends on factors such as temperature, pressure, and the properties of the diffusing species and the medium
• Higher temperatures lead to increased molecular motion and higher diffusivities (molecules have more kinetic energy to overcome resistive forces)
• Smaller and more spherical molecules generally have higher diffusivities due to reduced interactions with the medium and other molecules

Medium properties and multi-component systems

• The properties of the medium through which diffusion occurs, such as viscosity and porosity, can impact the diffusivity and the overall diffusion rate
• Higher viscosity media offer more resistance to molecular motion, reducing diffusivity (molecules have to overcome more drag forces)
• In porous media, the effective diffusivity is lower than in free space due to the tortuous path molecules must follow (diffusion path is longer than the straight-line distance)
• In multi-component systems, the presence of other species can influence the diffusion of a particular component through interactions (attractive or repulsive forces) and competition for space
• Diffusion in multi-component systems can be described by the Maxwell-Stefan equations, which account for the interactions between different species

Steady-state vs unsteady-state diffusion

Solving steady-state diffusion problems

• Steady-state diffusion problems involve situations where the concentration gradient remains constant over time, resulting in a constant diffusive flux
• Steady-state problems can often be solved using Fick's first law, as the diffusive flux is independent of time
• In one-dimensional steady-state diffusion, the diffusive flux is given by $J = -D(ΔC/Δx)$, where $ΔC$ is the concentration difference and $Δx$ is the diffusion distance
• Examples of steady-state diffusion include the diffusion of gases through a porous membrane (constant concentrations on both sides) and the diffusion of heat through a wall (constant temperatures on both sides)

Solving unsteady-state diffusion problems

• Unsteady-state diffusion problems involve time-dependent concentration gradients and require the use of Fick's second law
• Solving unsteady-state problems often involves partial differential equations and may require numerical methods or simplifying assumptions, such as constant diffusivity or semi-infinite media
• Initial and boundary conditions are crucial in solving unsteady-state problems, as they define the concentration distribution at the start and the behavior at the system boundaries
• Examples of unsteady-state diffusion include the diffusion of a drug from a tablet into the body (concentration decreases over time) and the diffusion of heat into a solid during transient heating (temperature changes over time)
• Analytical solutions to unsteady-state diffusion problems exist for simple geometries and boundary conditions, such as the diffusion in a semi-infinite medium with a constant surface concentration (error function solution)