4.1. Diffusion in Fluid Phase

In box models, we write the mass balance in the sediment volume as

Rate accumulation is non-zero and represents the movement of material.

This section explains how to set up a mass balance in sediment scenarios. The formula represents the change in concentration over time within a defined volume of sediment, considering the inflow and outflow of materials. The terms $F_{in}$ and $F_{out}$ represent the rates at which material enters and exits the volume, respectively.

Imagine a bathtub filling with water. The water flowing in is comparable to $F_{in}$, while the water draining out corresponds to $F_{out}$. If more water comes in than goes out, the water level (concentration in the sediment) will rise, similar to how material accumulates in sediments.

The movement into and out of the system occurs only by diffusion, given by the equation:

This indicates a concentration gradient driving the diffusion process.

This chunk emphasizes that diffusion is the only mechanism interacting with the material in the fluid phase. The equation indicates that the rate of flow in (or out) is proportional to the negative concentration gradient, which means diffusion occurs from areas of high concentration to areas of low concentration, driving materials into or out of the sediment.

Consider a drop of food coloring in a glass of water. Initially, the color is concentrated where it was dropped. Over time, it spreads out, moving from the high concentration area (the drop) to low concentration areas (the rest of the water) through diffusion.

In porous media, the effective diffusivity $D_{eff}$ is defined as:

$\phi$ is the porosity, affecting how diffusion behaves in such materials.

This section describes how diffusion is impacted by the structure of the porous medium. The effective diffusivity adjusts the basic diffusion coefficient based on the porosity of the material, indicating how much of the space is occupied by voids through which diffusion can occur. A higher porosity means more space for diffusion, increasing the effective diffusivity.

Think about moving through a crowded room vs. an empty hallway. In the hallway (high porosity), you can move quickly (high effective diffusivity). In the crowded room (low porosity), your movement is restricted, slowing you down (low effective diffusivity).

The local equilibrium assumption implies that there is a rapid exchange between solid and fluid, with:

Where $K_d$ is the partition coefficient relating concentrations in the solid and the liquid.

In this context, the local equilibrium assumption simplifies the modeling by asserting that the rate at which adsorption occurs on the solid is almost equal to the rate of diffusion into the porosity. This assumption allows for a direct relationship between concentrations in the solid phase and the fluid phase, which streamlines calculations and predictions in sediment analysis.

Consider a sponge soaking up water. If you dip a sponge in water, it quickly absorbs it, reaching a balance of water inside (solid phase) and that outside (fluid phase). Here, the water inside the sponge is proportional to the amount of water in the surrounding environment, demonstrating a local equilibrium.

The retardation factor, defined as:

indicates how much slower substances diffuse in the presence of solids.

The retardation factor highlights an important concept in diffusion: when solids adsorb substances, it can slow down their movement. High levels of solids or high partition coefficients lead to a larger retardation factor, meaning diffusion happens at a slower rate due to the interaction with solids. This is critical in environmental scenarios where pollutants may be retained in sediments.

Imagine pouring syrup into a glass of water. The syrup moves slowly because it interacts with the molecules of water, similar to how a pollutant interacts with sediments. The more syrup you pour (or solids present), the longer it takes for the liquid to diffuse throughout the glass.

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