Non-Linear Partitioning Example: The Langmuir Isotherm

In many systems, species The chemical (or non-chemical, such as bacterial or viral) constituents that are stored and transported through an environmental system in a contaminant transport model. In GoldSim, the Species element defines all of the contaminant species being simulated (and their properties). partition between media Materials (such as water, sand, clay, air) that constitute (are contained within) transport pathways. GoldSim provides two types of elements for defining media: Fluids and Solids. (e.g., water and sediment) according to a non-linear isotherm, in which the degree of partitioning varies with concentration. To illustrate the way in which you could represent non-linear partitioning, we will consider one common type of non-linear isotherm, the Langmuir isotherm. This particular example file, Langmuir.gsm, can be found in the Contaminant Transport Examples folder in your GoldSim directory (accessed by selecting File | Open Example... from the main menu).

The Langmuir isotherm is described mathematically as follows:

where:

csorbed is the sorbed concentration (M/M);
csorbed,max is the maximum sorbed concentration (M/M);
cdiss is the dissolved concentration (M/L3); and
KLang is the Langmuir partition constant (L3/M).

This equation can be rearranged to yield an effective solid/water partition coefficient:

To illustrate this, consider and example in which we simulate a single Cell containing 1 m3 of Water and 1kg of Sand.

The Cell contains an initial mass of Species1. The Cell has no mass flux links; our objective is to simply plot the concentration of Species1 in the Sand as a function of the total mass in the Cell. We do this by running the simulation for multiple realizations and varying the initial mass in the Cell (a uniform distribution between 0 and 20 g). We will assume that the maximum sorbed concentration is 1 g/kg, and the Langmuir partition constant is 3 m3/kg.

To simulate this system in GoldSim, you would do the following:

  1. Define a single species and two media (Water and Sand);
  2. Define a single Cell containing 1 m3 of Water and 1 kg of Sand;
  3. Define the initial mass in the Cell as a Stochastic (a uniform between 0 and 20 g);
  4. Create two constants, the maximum sorbed concentration and the Langmuir partition constant;
  5. Define the partition coefficient for Species1 for the Cell using the equation specified above; and
  6. Specify the simulation settings (in this case 100 realizations with 10 timesteps for 1 day) and run the model.

Note that step five creates a recursive model (since the dissolved concentration is a function of the partition coefficient, and the partition coefficient is a function of the dissolved concentration). Hence, when you do this, you must utilize a Previous Value element An element that outputs the value of its input from the previous model update..

The output of this simulation, in the form of a scatter plot of the sorbed concentration (the concentration in Sand) versus the dissolved concentration, is shown below:

For linear partitioning, this would be a straight line. As can be seen, non-linear isotherms "bend" over and approach a maximum.

Note: Since you can only reference the concentration in the fluidleaving a Pipe or Aquifer pathway A transport pathway element that is intended to represent a feature that essentially behaves as a fluid conduit. Internally, an Aquifer pathway actually performs its computations by creating a temporary set of linked Cell elements during the simulation., it will likely not be appropriate to use this to compute the partition coefficient throughout the entire length of the pathway. Hence, in general, non-linear partition coefficients The ratio of the species’ concentration in a medium to its concentration in the Reference Fluid at equilibrium. Partition coefficients are inputs to Solid and Fluid elements. should only be applied (i.e., defined for solids present) within Cell pathways.