# Scaling Tendencies

## Contents |

## What is Scaling?

Scaling is the deposition of a mineral salt on processing equipment. Scaling is a result of supersaturation of mineral ions in the process fluid. Through changes in temperature, or solvent evaporation or degasification, the concentration of salts may exceed the saturation, leading to a precipitation of solids (usually crystals).

To understand Scaling, the Scaling Tendency and Scaling Index definitions are used. These definitions depended on the Solubility Product Constant, K_{sp} and Ion Activity Product, IAP definitions.

**Solubility Product Constant, K _{sp}**

The solubility of ionic compounds of salts and minerals in water are governed by a solubility equilibrium expression and a solubility product constant known as K_{sp}. It is important to note that the solubility product, K_{sp} is a function of both temperature and pressure. Consider the general dissolution reaction below (in aqueous solutions):

aA_{(s)}⇌bB_{(aq)}+dD_{(aq)}

With equilibrium constant K_{sp} defined as:

K_{sp}=(*a*_{B})^{b}∙(*a*_{D})^{d}

Where, *a*_{B} and *a*_{D} are the activities of the aqueous species. The activity of any species i is defined as the product of its concentration in molality by its corresponding activity coefficient:

*a*_{i}=m_{i} γ_{i}

**Ion Activity Product, IAP**

A real solution may not be in the state of equilibrium. This non-equilibrium state is described by the ion activity product (IAP). It has the same form as the equilibrium constant K_{sp}, but involves the actual activities of the species in solution.

IAP=(*a*_{B})^{b}_{actual}∙(*a*_{D})^{d}_{actual}

## Scaling Tendency and Scaling Index

The Scaling Tendency (ST) is defined as the ratio of the Ion Activity Product (IAP) divided by the equilibrium constant (K_{sp}).

**Equation (1)** ST=IAP/K_{sp}

Scaling tendencies are essentially saturation ratios. Thus, if

- ST < 1 Indicates sub-saturation, and the solid is not expected to form
- ST = 1 Indicates saturation, and the solid is in equilibrium with water
- ST > 1 Indicates supersaturation, and solids will form

**Note:** The Scaling Tendency (ST) is reported in the software as Post-Scale.

The Scale Index (SI) (aka: Saturation Index in the literature), is given by the following relationship:

**Equation (2)** SI =log_{10}(IAP/K_{sp})

Thus, if

- SI < 0 Indicates sub-saturation, and the solid is not expected to form
- SI = 0 Indicates saturation, and the solid is in equilibrium with water
- SI > 0 Indicates supersaturation, and solids will form

**Note:** The Scaling Index (SI) is reported in the software as SI, Index

## Pre-Scaling Tendency and Scaling Index

Pre-Scaling tendency is defined as the scaling tendency before any solids are formed (this can be seen as all the species suspended in solution). The same formulas for ST and SI are applied (Equations 1 and 2).

The Pre-Scaling tendency is reported in the software as Pre-Scale.

**Note:** Many industries, notably the up-stream oil & gas industry, use the pre-scaling tendency to make design decisions about adding anti-scaling and anti-fouling agents or if the asset is as risk.

## Difference between Post-scale and Pre-scale

- Pre-Scale: The saturation ratio before solids precipitate.
- Post-Scale: The saturation ratio AFTER solids precipitate (if solids are selected).
- S, ST – Saturation, Scale Tendency: The ratio of the concentration (activity) to its solubility (S=1).
- SI – Scale Index: Log(ST).

**Scaling Tendency** is the saturation ratio after all potential solids come to equilibrium with water. This is the true equilibrium condition (time=∞). I

**Pre-scaling Tendency** represents the condition before any solids are allowed to form. This is a non-equilibrium condition and can be viewed as the condition where time=0.

## Calculating Scaling Tendency: An Example

Below there is an example of how the software calculates the Scaling Tendency. Consider the dissolution of sodium bicarbonate:

NaHCO_{3(s)} = Na^{+} + HCO_{3}^{-}

Scaling Tendency is defined as ST = IAP/K_{sp}

Where **IAP** is the **Ion Activity Product**, and is defined as the product of specific ions (in this case the ions resulting from the dissociation of a particular solid); and **K _{sp}** is the

**Solubility Product Constant**which gives the thermodynamic limit of ion availability.

Let’s consider a 1.0 molal NaHCO_{3} solution. The IAP is calculated as follows:

IAP = [gamma_{Na+}]*[m_{Na+}]*[gamma_{HCO3-}]*[m_{HCO3-}]

Assuming Ideal Solution Activities:

gamma_{Na+} = 1.0

gamma_{HCO3-} = 1.0

m_{Na+} = 1.0

m_{HCO3-} = 1.0

Then,

IAP = (1.0)(1.0)(1.0)(1.0)

IAP = 1.0

Defining the Ksp

K_{sp} = [gamma_{Na+}]*[m_{Na+}]*[gamma_{HCO3-}]*[m_{HCO3-}]

K_{sp} = 0.403780

The Scaling Tendency is then the ratio of available ions to the thermodynamic limit:

ST = IAP/K_{sp}

ST = 1.0/0.403780

ST = 2.48

Was assuming ideal conditions valid?? The actual species concentration and activity coefficients (calculated by the OLI software) are:

gamma_{Na+} = 0.598

gamma_{HCO3-} = 0.596

m_{Na+} = 0.894

m_{HCO3-} = 0.866

This results in a different IAP:

IAP= (0.598)(0.894)(0.596)(0.866)

IAP=0.276

The new Scaling Tendency is the following:

ST = IAP/K_{sp}

ST = 0.276/0.40378

ST = 0.683