Simulating Solubility Limits in Cells

One of the most powerful features of Cell pathways is that you can impose a solubility constraint within them. Solubility limits are defined for each species within Fluid elements.

The solubility represents the maximum dissolved concentration that the species can have within a Fluid in a Cell.

Note: If a Fluid contains suspended particulates, the effective concentration could exceed the solubility limit, since the effective concentration includes the effect of the species mass associated with the suspended solids. In fact, for low solubility species, the species mass associated with suspended Solids could be the most important contributor to the effective concentration.

When a solubility constraint is applied for a species in a Cell, the Cell has a saturation capacity with respect to that species, which represents the maximum amount of species mass the Cell can contain before the species mass will start to precipitate out of solution.

The saturation capacity of a cell is a function of the solubility limit, the partition coefficients for the other media present in the cell, and the quantities of the media present in the cell:

If the total species mass in the Cell does not exceed the saturation capacity, the solubility limit has not been reached, and concentrations in the various media in the Cell are computed based on partition coefficients and media quantities.

If, however, the saturation capacity is exceeded, media concentrations are computed differently. In particular, the concentration in the Reference Fluid (if it is present) is set to the solubility limit, and the concentrations in other media are computed based on their partition coefficients and the concentration in the Reference Fluid. In addition, the concentrations in Solids in the Cell are incremented by the amount of "precipitated" species mass (the difference between the total mass in the Cell and the saturation capacity).

• The concentration in the Reference Fluid is set to the solubility limit for species i, sol_i [M/L³];

where K_fi is the partition coefficient for the species in fluid f [dimensionless].

• The concentration [M/M] in suspended (particulate) Solids (c_pi) is computed as:

where K_pi is the partition coefficient for the species in suspended Solid p[L³/M].

• The concentration [M/M] in Solids which are not suspended (c_si) is computed as:

where K_si is the partition coefficient for the species in Solid s[L³/M], mtot_i is the total mass in the cell [M], msat_i is the saturation capacity [M], VM_s is the mass of (non-suspended) Solid in the Cell, and NS is the number of non-suspended Solids in the Cell.

Note the difference between the treatment of suspended Solids and those that are not suspended. In particular, for computational reasons, the mass in the cell in excess of the saturation capacity (the "precipitated" mass) is assumed to be precipitated only onto those Solids which are not suspended. Hence, although species mass can be partitioned onto (and transported with) particulate Solids, precipitated species mass is never associated with suspended Solids.

Note: A medium that has zero volume (Fluids) or mass (Solids) cannot have a species concentration. Any mass in the Cell will partition into other Fluids or Solids and precipitated mass will become associated with any (non-suspended) Solids present in the Cell (proportional to the mass of each Solid).

Note: GoldSim provides a special mass flux link that allows you to selectively remove precipitated mass from a Cell.

If you are simulating multiple isotopes of the same element, GoldSim must handle the solubility calculations in a special way. This is because, physically, 1) solubility is actually an elemental property (as opposed to an isotopic property); and 2) the solubility is "shared" amongst the isotopes.

1. If solubilities are defined by species, it assumes that the solubility entered for the first isotope of the element in the species list is the elemental solubility (and if the solubility is specified by mass concentration, it is converted to molar concentration based on the specified Weight of that first isotope);

2. It uses the sum of the moles of all isotopes of an element present to determine if the elemental molar saturation capacity has been exceeded; and

3. If the elemental saturation capacity has been exceeded, it "shares" the elemental solubility in the Reference Fluid among the isotopes of the element based on the isotopic molar ratios. (For example, if the uranium isotopes ²³⁵U and ²³⁸U were present in a Cell at a ratio of 2 moles to 1 mole, and the total moles of uranium in the Cell exceeded the molar saturation capacity, the dissolved molar concentration of ²³⁵U would be computed as 2/3 of the elemental uranium molar solubility and the dissolved molar concentration of ²³⁸U would be computed as 1/3 of the elemental uranium molar solubility.)

Note that this “sharing” of the solubility amongst isotopes results in the governing equations becoming non-linear (and hence more difficult to solve).

Warning: Solubility limits are not applied in Cells with a Defined Concentration boundary condition. If the Defined Concentration for a species is above the solubility limit, the solubility limit is ignored, the Defined Concentration is applied, and a warning is written to the Run Log.

Warning: The “turnover rate” for a species in a Cell is defined as the ratio of the discharge rate for the species to the current amount of species in the Cell. The product of the turnover rate and the timestep indicates how many times the Cell is “flushed” within one timestep. If this product is very high, due to round-off error the species could oscillate between a saturated and unsaturated condition, which can cause the solution to slow down considerably. To prevent this, if the turnover rate multiplied by the timestep exceeds 1000 (implying that in one timestep, the Cell flushes its mass on the order of 1000 times), solubility limits are no longer applied (for all species). The fact that this occurs is written to the Run Log (but it is not treated as a Warning or Error, so that you will not be notified). Note that if the turnover rate varies dramatically, application of solubility limits may be activated and deactivated multiple times during a simulation. However, only the first instance is written to the Run Log.

msat_i	=	saturation capacity for species i in the cell [M];
sol_i	=	solubility of species i in the Reference Fluid [M/L³];
K_gi	=	partition coefficient between medium g and Reference Fluid for species i [(L³ /L³) for Fluids or (L³/M) for Solids];
VM_g	=	quantity (volume or mass) of medium g in cell [L³ for Fluids or M for Solids]; and
NM	=	the number of media in cell.