3.1. Flux Equation and Error Functions

We were talking about contaminant transport in sediments. So, last time we looked at a very simple case where the contaminant is uniform. Our model here is the pore water and this is w in the sediment. The initial contamination in the sediment V = V0 is uniform. Last time we mentioned the semi-infinite system where at time t = 0 and all z, V = V0, where V is the concentration of A in pore water.

In studies of contaminant transport within sediments, we start with a scenario where the contaminant is uniformly distributed at the initial time, t = 0. This setup is termed as a 'semi-infinite system,' meaning we consider a vast or infinite body of sediment where the contaminant exists at a uniform concentration (V0) in the pore water across all depths (z). This is a theoretical simplification that helps us to apply mathematical models effectively.

Imagine a sponge soaked in water. If the sponge represents sediment and the water represents contaminant concentration, when you first submerge the sponge in clean water, the entire sponge's water content is uniform. As time passes, if you pull that sponge out, the water starts to drip out unevenly, just like how the contaminant concentration changes over time. Initially, it's uniform, but as the sponge dries (or the contaminant is removed), some areas will lose water faster than others.

We also have a boundary condition at z = 0, which represents the interface where mass transfer occurs. There, we stated that the material is coming to the interface by diffusion and being carried away via convective actions.

At the z = 0 boundary, which is the interface between the sediment and the water, we observe two main behaviors: diffusion, where contaminants move from areas of high concentration to low concentration (inside the sediment), and convection, where moving water can carry away the contaminants at the interface. This boundary condition acknowledges the complexity of both processes and defines how we calculate the mass transport through this interface.

Think of it like a busy kitchen where chefs are cooking food (sediment) and serving it. As the chefs prepare the dishes (contaminants), they serve them to customers waiting at the counter (the interface). The way the traffic of customers at the counter (convection) influences how quickly the food is served is similar to how contaminants are carried away from the sediment to the water. If there's a lot of demand (fast convection), the food will disappear quickly from the countertop, even if the chefs are cooking slowly (diffusion).

The flux is given as the function of time, and we discussed the flux in terms of the error function. The error function (erf) is defined, and its complementary (erfc) helps in calculations related to diffusion processes.

The flux of contaminants from sediment to water over time is modelled using the error function, erf(x), which mathematically represents how substances disperse through a medium over time. These functions are critical in solving differential equations that describe diffusion, allowing for precise calculations regarding the movement of contaminants. In our analysis, as time progresses, the relationship between flux and concentration becomes refined mathematically through these functions.

Consider throwing a pebble into a calm pond. The ripples spreading outward represent how a contaminant disperses in water over time. Initially, the ripples are concentrated near where the pebble landed, similar to high flux at the start. As they spread out, the intensity diminishes, akin to the error function demonstrating how rapidly contaminant concentration decreases as it diffuses away in the water. Just like ripples, the behavior of the error function shows how things spread out over time.

As time increases, we expect the flux to decrease, meaning the concentration of contaminant available at the surface also decreases.

Over time, as contaminants diffuse out of the sediment, the concentration at the surface diminishes. This is a typical scenario in environmental systems where pollutants are constantly being diluted or depleted. It suggests that while initially there may be a high concentration of pollutants available, their availability reduces as more time elapses, leading to a decrease in the flux observed at the sediment-water interface.

Think about a bath filled with warm water. Initially, when you first add cold water, the temperature (representing contaminant concentration) is significantly affected because the cold water is being introduced. Over time, as the two temperatures mix, the immediate influence of the cold water (like a contaminant) diminishes, leading to a more uniform temperature (lower concentration) throughout the bath. This reflects how, over time, the amount of contaminant availability decreases at the surface.

We discussed different boundary conditions to simplify our calculations such as assuming zero concentration at the interface to streamline solving the equations.

To facilitate the mathematical modelling of flux, we can simplify our assumptions about concentrations at boundaries. By assuming that concentration at the interface is effectively zero, we reduce complexity in our calculations, allowing for a more straightforward application of models. It's an assumption that, while not physically exact, helps in understanding the rate of contaminant transfer.

Imagine a factory's conveyor belt system. If you assume the belt always starts empty (zero concentration) when you introduce a product, you can more easily calculate how many products will be on the belt after a certain time. This simplification allows for easier planning and management, even though in reality, there may always be some product residuals left on the belt.

--