4. Diffusion

So what we are doing? We do this normally in all box models kind of scenarios. We write again the mass balance in the sediment volume. This is delta x, delta y, delta z is the differential volume of the system. We write

∂C/∂t = (Rate_in - Rate_out) / V.

Now, here I can write other terms if I want to, yeah, I can write many things. Same as box model, I can write whatever is happening inside the system.

This chunk introduces the concept of mass balance, which is crucial in understanding the dynamics of sediment systems. The mass balance equation essentially states that the change in concentration (∂C/∂t) within a given volume is equal to the difference between the rate of material entering and leaving that volume. The variables delta x, delta y, and delta z represent small changes in space dimensions, defining a tiny volume in which we analyze the mass flow. This framework is similar to what is used in simpler models, but it accounts for complex interactions in the sediment.

Imagine a bathtub where some water is pouring in at the same rate as water is draining out. The mass balance equation here would help us figure out how the water level changes over time based on how much is coming in versus how much is going out. In sediments, it's like determining how pollutants accumulate based on how they enter and leave a soil volume.

The rate of accumulation is now, it is nonzero, it is not steady state, something is not going out in the same manner. So we have to write this in terms of diffusion. This is only by diffusion, so which means it is

∂C/∂t = D_A ∇^2C.

In this chunk, the professor points out that unlike static systems, sediment systems experience a nonzero rate of accumulation. This means concentrations can change over time due to input from several sources, not just balance out from import and export. The diffusion term (D_A ∇^2C) describes how materials, such as pollutants, spread through the sediment, indicating that concentration changes are also driven by spreading out from areas of high concentration to areas of low concentration.

Think of how a drop of food coloring disperses in water. At first, it’s concentrated in one spot, but over time, it spreads out until the color is uniform. Similarly, in sediments, if a pollutant is introduced, it will initially have a high concentration in one area, but over time will diffuse through the sediment, affecting larger areas.

This is called as effective diffusivity in a porous medium and it is a function of the porosity.

D_A = D_f * (Φ^n).

Effective diffusivity refers to how well a substance can move through a porous medium (like soil) as opposed to through free fluid. It highlights that when dealing with porous materials, the presence of structures (like soil particles) can impede movement, making diffusion less efficient. The porosity (Φ) and an exponent (n) adjust the diffusivity to account for how much space is available for movement.

Consider a sponge submerged in water. The water can easily move into the spaces between the sponge's fibers, but it takes longer for it to completely saturate the sponge. The sponge's structure limits how freely the water can spread, similar to how soil affects pollutant diffusion.

Local equilibrium assumption assumes that when a solid is near a fluid, the rate at which diffusion is occurring is almost the same order of magnitude at which this adsorption is happening.

This assumption simplifies the complex interactions between solids (such as soil particles) and fluids (like water carrying pollutants) by saying that adsorption and diffusion happen simultaneously and at comparable rates. This means that as contaminants are diffusing into the solid matrix of sediment, they are also getting adsorbed (sticking) onto the solid particles at a similar pace.

Imagine a sponge while you're pouring water over it. The water seeps in at nearly the same rate as it absorbs into the sponge's fibers. The local equilibrium assumption works under the premise that this balance remains constant, reflecting how pollutants interact with sediment.

This term in the denominator is known as a retardation factor, which is equal to (ρ_s + K_d * ρ_w).

The retardation factor explains how factors such as adsorption to solids (denoted by solid density ρ_s and distribution coefficient K_d) slow down the rate of diffusion. A large retardation factor indicates that the presence of solids substantially impedes the movement of contaminants through the water-soil interface.

Think of a crowded hallway where people (representing pollutants) are trying to get from one end to the other. If the pathway is cluttered with obstacles (the soil particles acting as adsorbents), the people will move through the hallway much slower compared to a clear, unobstructed path. The retardation factor quantifies this slowdown caused by the solid structures.

--