Non-Linear Partitioning Example: The Langmuir Isotherm

In many systems, species partition between media (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

= the sorbed concentration (M/M);

csorbed,max

= the maximum sorbed concentration (M/M);

cdiss

= the dissolved concentration (M/L3); and

KLang

= 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.  

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:

A screenshot of a map Description automatically generated

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 fluid leaving a Pipe or Aquifer pathway, 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 should only be applied (i.e., defined for solids present) within Cell pathways.