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Computerized Ion Association Model Profiles Complete Range of Cooling System Parameters

Robert J. Ferguson, French Creek Software, Inc., Kimberton, PA 19442 U.S.A.

Paper Number IWC-91-47

OCTOBER 21 - 23, 1991

Keywords: cooling water, indices, scale, water chemistry, computer models, ion association

This paper describes the application of a microcomputer (PC) based software system to establishing operating parameter limits for open recirculating cooling systems. The software used evaluates recirculating water chemistry over the typical, user defined, operating range of concentration ratio, temperature, and pH to provide an in-depth, panoramic estimation of deposition potentials. The deposition potentials calculated for a cooling water are based upon the free concentration of reactants and account for common ion effects. The approach of profiling the entire operating range using an ion association model overcomes two criticisms of water chemistry evaluations and their applicability towards predicting scale problems in a system and comparing indices versus results between cooling systems.

Many cooling water chemistry evaluations are based upon a single water analysis. Even sophisticated computerized programs in use rely upon a single recirculating water analysis and a single set of operating parameters for an evaluation of scale potential and corrosivity. Their evaluations and predicted deposition potentials are based, in most cases, upon a single pH and single temperature. Even small changes in water chemistry or operating parameters can invalidate the results of single point evaluations. In the worst case, evaluation at one set of conditions (e.g. highest temperature, highest pH) can effectively portray scale potential for a foulant such as calcium carbonate under the harshest conditions for it, but result in the evaluation of a foulant such as amorphous silica under the conditions where it is most soluble.

Commonly used indices for scale potential base their calculations upon the analytical values for scale forming species. Common ion effects are not included in the calculations.1 This can result in decisions, such as the maximum concentration ratio for operation, based upon inflated scale potentials.

The program discussed uses an ion association model to estimate the saturation level (ion activity product over solubility product) for common foulants including calcium carbonate, calcium sulfate, amorphous silica, calcium fluoride, magnesium silicates, and tricalcium phosphate. Saturation level calculations include ion pairing to account for common ion effects and provide a more reproducible, and representative driving force. Traditional saturation derived indices (Langelier2, Ryznar3 and Puckorius4) are also profiled by the system based upon the analytical values input.

Examples are presented which compare the indices calculated by traditional methods to those done using an in-depth, panoramic profile over the typical operating range for a cooling system.

THE CONCEPT OF SATURATION - A majority of the indices used routinely by water treatment chemists are derived from the basic concept of saturation. A water is said to be saturated with a compound (e.g. calcium carbonate) if it will not precipitate the compound and it will not dissolve any of the solid phase of the compound when left undisturbed, under the same conditions, for an infinite period of time. A water which will not precipitate or dissolve a compound is at equilibrium for the particular compound.

By definition, the amount of a chemical compound which can be dissolved in a water and remain in solution for this infinite period of time is described by the solubility product (Ksp). In the case of calcium carbonate, solubility is defined by the relationship:

              (Ca)(CO3) = Ksp


  • (Ca) is the activity of calcium (CO3) is the carbonate activity
  • Ksp is the solubility product for calcium carbonate at the temperature under study.

In a more generalized sense, the term (Ca)(CO3) can be called the Ion Activity Product (IAP) and the equilibrium condition described by the relationship:

                IAP = Ksp

It can be shown that the Langelier Saturation Index is the base ten logarithm of calcite saturation level based upon total calcium in the water, an estimate of carbonate calculated from total alkalinity, and the solubility product for the calcite polymorph of calcium carbonate.2,5

The degree of saturation of a water is described by the relationship of the ion activity product (IAP) to the solubility product (Ksp) for the compound as follows:

  • If a water is undersaturated with a compound: IAP< Ksp
    (It will tend to dissolve the compound).

  • If a water is at equilibrium with a compound: IAP= Ksp
    (It will not tend to dissolve or precipitate the compound).
  • If a water is supersaturated with a compound: IAP>Ksp
    (It will tend to precipitate the compound).

The index called Saturation Level, Degree of Supersaturation, or Saturation Index, describes the relative degree of saturation as a ratio of the ion activity product (IAP) to the solubility product (Ksp):

            Saturation Level = _______
            ____________ ____ Ksp

In actual practice, the saturation levels calculated by the various computer programs available differ in the method they use for estimating the activity coefficients used in the IAP; they differ in the choice of solubility products and their variation with temperature; and they differ in the dissociation constants used to estimate the concentration of reactants (e.g. CO3 from analytical values for alkalinity, PO4 from analytical orthophosphate).

Table 1 defines the saturation level for common cooling water foulants and provides the basis for discussion of these foulants in this paper.

ION PAIRING - The Saturation Index discussed can be calculated based upon total analytical values for the reactants. Ions in water, however, do not tend to exist totally as free ions.6,7,8 Calcium, for example, may be paired with sulfate, bicarbonate, carbonate, phosphate and other species. Bound ions are not readily available for scale formation. The computer program calculates saturation levels based upon the free concentrations of ions in a water rather than the total anaytical value which includes those which are bound.

Early indices such as the Langelier Saturation Index (LSI) for calcium carbonate scale, are based upon total analytical values rather than free species primarily due to the intense calculation requirements for determining the distribution of species in a water. Speciation of a water is time prohibited without the use of a computer for the iterative number crunching required. The process is iterative and involves:

  1. Checking the water for a electroneutrality via a cation-anion balance, and balancing with an appropriate ion (e.g sodium or potassium for cation deficient waters, sulfate, chloride, or nitrate for anion deficient waters).
  2. Estimating ionic strength, calculating and correcting activity coefficients and dissociation constants for temperature, correcting alkalinity for non-carbonate alkalinity.
  3. Iteratively calculating the distribution of species in the water from dissociation constants
    (a partial listing is outlined in figure 1).
  4. Checking the water for balance and adjusting ion concentrations to agree with analytical values.
  5. Repeating the process until corrections are insignificant.
  6. Calclulating saturation levels based upon the free concentrations of ions estimated using the ion association model (ion pairing).

The use of ion pairing to estimate the free concentrations of reactants overcomes several of the major shortcomings of traditional indices. Indices such as the LSI correct activity coefficients for ionic strength based upon the total dissolved solids. They do not account for "common ion" effects.1 Common ion effects increase the apparent solubility of a compound by reducing the concentration of reactants available. A common example is sulfate reducing the available calcium in a water and increasing the apparent solubility of calcium carbonate. The use of indices which do not account for ion pairing can be misleading when comparing waters where the TDS is composed of ions which pair with the reactants versus ions which have less interaction with them.

When indices are used to establish operating limits such as maximum concentration ratio or maximum pH, the differences between the use of indices calculated using ion pairing can be of extreme economic significance. In the best case, a system is not operated at as high a concentration ratio as possible, because the use of indices based upon total analytical values resulted in high estimates of the driving force for a scalant. In the worst case, the use of indices based upon total ions present can result in the establishment of operating limits too high. This can occur when experience on a system with high TDS water is translated to a system operating with a lower TDS water. The high indices which were found acceptable in the high TDS water may be unrealistic when translated to a water where ion pairing is less significant in reducing the apparent driving force for scale formation.

Table 2 summarizes the impact of TDS upon Langelier Saturation Index when calculated using total analytical values for calcium and alkalinity, and when calculated using the free calcium and carbonate concentrations calculated using an ion association model. The same data is presented graphically in Figure 2.

ECONOMIC IMPACT OF ION PAIRING - Indices based upon ion association models provide a common denominator for comparing results between systems. For example, calcite saturation level calculated using free calcium and carbonate concentrations, has been used successfully as the basis for developing models which describe the minimum effective scale inhibitor dosage which will maintain clean heat transfer surfaces.9,10,11,12 The calcite saturation level driving force provided a common denominator for applying the models to cooling systems across the country with varying water quality, and varying degrees of common ion effects mediating the apparent driving force for scale formation. Previous models, based upon total analytical values, did not provide results as reproducible between systems.

Indices based upon ion pairing provide an excellent basis for optimising operating parameters for the high alkalinity, high pH all organic treatment programs in use today. Many of these programs rely upon phosphonates for calcium carbonate scale control. It has been reported, (and is the author's experience), that most inhibitor programs lose scale control when the Langelier Saturation Index exceeds 2.5 .13 A magic index value is used by most water treatment companies in establishing an upper limit for concentration ratio and pH to prevent loss of control. This value is based upon laboratory and field experience with the scale inhibitors. The use of total analytical concentrations for calcium and alkalinity can result in the establishment of limits significantly lower than possible, if the limits are based upon a lower TDS water than that of the cooling system, or higher than appropropriate, if established with data from a higher TDS water. Indices calculated using ion pairing and free species provide a common denominator for comparing results between systems. Limits based upon indices calculated using total analytical concentrations are best applied to similar waters.

Calcite saturation level has also been used to establish the upper limit for all organic programs. A typical "magic" number of 150 is reasonable for free ion calcite saturation level.14

This is comparable to a Langelier Saturation Index of 2.50 in waters with makeup compositions similar to the Great Lakes. Figure 3 and Figure 4 compare the "magic number" concentration ratio for a low and high sulfate water based upon an LSI of 2.5 and a free ion calcite saturation level of 150.

In the case of the low sulfate, lower TDS water, the LSI limit of 2.50 and the calcite saturation level limit of 150 both occur near a concentration ratio of 5.6 to 5.7. The difference between the two limits is negligible.

In the case of the high sulfate water, the use of the ion association model to determine the maximum concentration ratio would increase the maximum from 6.5 to7.8 in comparison to the concentration ratio limit which would be imposed based upon the use of the traditional Langelier Saturation Index for the calculations.

The differences in limits between the two waters result from the impact of sulfate upon the free calcium ion concentration. Table 3 compares the concentration ratio limit which would be calculated based upon the LSI with limits based upon ion association model calcite saturation level for four (4) waters.

It should be noted that these waters represent a computer simulation and were chosen to demonstrate a trend. The addition of other anions and cations to the analysis will further impact the trends, and have varying impacts upon the concentration ratios where the LSI or calcite saturation level limits are reached.

The use of ion association model indices can provide a common denominator for the water treater. When faced with a variety of water sources and operating parameters, ion pairing provides a more reproducible driving force for the development of operational limits for inhibitor programs than indices calculated using total anaytical values. The use of free ions for index calculations eliminates or minimizes the impact of total dissolved solids and their composition upon the calculated index to allow for better reproducibility between waters and cooling systems.

OPERATING RANGE SOLUBILITIES - Many cooling water evaluations assume that the cooling system is static. Indices for scale potential are calculated at the "harshest" conditions for the foulant under study. In the case of calcium carbonate scale, indices are typically calculated at the highest expected temperature and highest expected pH: the conditions where calcium carbonate is least soluble. In the case of silica, the opposite conditions are used. Amorphous silica has its lowest solubility at the lowest temperature, and lowest pH encountered. Indices calculated under these conditions would be acceptable in many cases. Unfortunately, cooling systems are not static. This section describes the use of operating range profiles to answer questions such as:

  • What happens if the pH rises a tenth or two above the control range maximum?
  • What happens with acid overfeed and a pH well below the minimum desired?
  • What happens if the system cycles above the maximum target concentration ratio?

The foulants silica and tricalcium phosphate are used as examples to demonstrate the use of operating range profiles in developing an in-depth evaluation of scale potential, and the impact of loss of control.

Silica - Guidelines for the upper silica operating limits have been well defined in water treatment practice, and have evolved with the treatment programs. In the days of acid chromate cooling system treatment, an upper limit of 150 ppm silica as SiO2 was common. The limit increased to 180 ppm with the advent of alkaline treatments and pH control limits up to 9.0 . Silica control levels approaching or exceeding 200 ppm as SiO2 have been reported for the current high pH, high alkalinity all organic treatment programs where pH is allowed to equilibrate at 9.0 or higher.

The evolution of silica control limits can be readily understood by reviewing the a silica solubility profile. As depicted in Figure 5, amorphous silica solubility increases with increasing pH. Silica solubility also increases with increasing temperature. In the pH range of 6.0 to 8.0 and temperature range of 70 to 90 °F, cooling water will be saturated with amorphous silica when the concentration reaches 106 ppm as SiO2 (70 °F), or 140 ppm (90 °F). These concentrations correspond to a saturation level of 1.0. The traditional silica limit for this pH range has been 150 ppm as SiO2. As outlined in Table 4, a limit of 150 ppm would correspond roughly to a saturation level of 1.4 at 70 F and 1.1 at 90 °F.

At the upper end of the cooling water pH range (9.0), silica solubility increases to 117 ppm (70 °F) and 140 ppm (90 °F). A control limit of 180 ppm would correspond to a saturation levels of 1.5 and 1.3, respectively. The author's experience is that a slight degree of saturation is acceptable in most systems, and that a silica saturation level of 1.1 to 1.2 at the cold well temperature is a conservative limit. In systems where concentration ratio is limited by silica solubility, it is recommended that the concentration ratio limit be re-established seasonally based amorphous silica saturation level or whenever significant temperature changes occur. Figure 6 profiles silica saturation level versus concentration ratio at 70 and 90 °F for a well water. An increase of concentration ratio from 4.3 to 6.1 is indicated based upon a target saturation level of 1.1.

Operating range profiles of silica saturation quickly provide a picture of pH and temperature limits within a system where silica is the limiting factor for operational concentration ratio. An overview such as the profile in figure 7 provide a point of reference for the degree of temperature, pH, or concentration change which indicates athat a review of the recirculating water chemistry is in order. Such profiles are also useful tools in establishing concentration ratio targets, and in determining if silica solubility is a limiting factor.

Cooling systems vary in the degree of supersaturation they can carry before measurable fouling occurs. As a result, it is recommended that saturation levels be used to establish limits based upon conditions where no silica deposition has been encountered. Once limits have been established, silica solubility profiles provide a useful tool in maximizing concentration ratio in silica limited systems.

Calcium Phosphate - Neutral phosphate programs can benefit from saturation level profiles for tricalcium phosphate. Treatment programs using orthophosphate as a corrosion inhibitor must operate in a narrow range of pH if satisfactory corrosion inhibition is to be achieved without catastrophic calcium phosphate deposition occurring. Operating range profiles for tricalcium phosphate can assist the water treatment chemist in establishing limits for pH, concentration ratio, and orthophosphate in the recirculating water. Such profiles are also useful in showing operators the impact of loss of pH control, chemical overfeed, or over concentration. Figure 8 is a typical profile for a neutral pH phosphate treatment program. It can be observed that tricalcium phosphate scale potential is negligible below a pH of 7.3 in this operating scheme. Minor alkaline pH excursions would quickly result in deposition if a copolymer or other calcium phosphate scale inhibitor were not present. The rapidity with which saturation level increases with pH is due to the fifth order nature of tricalcium phosphate (Figure 1). The saturation level is very sensitive to pH, which affects the orthophosphate concentration of the water, and calcium.

Frequent operating range profiles should be run for cooling systems which use pH control as the primary means for calcium phosphate fouling control.

SUMMARY - This paper presented an overview of the application of computerized modeling of cooling water scale potential. Prior to the advent of the AT and 386 based personal computers, ion association calculations were restricted to mainframe computers. As a result, few cooling water chemists were equipped to routinely profile the scale potential of a cooling system over the entire operating range. Indices were calculated based upon total analytical values for reactants such as calcium, and did not account for common ion effects. The use of ion association models allows the water treatment chemist the freedom to explore the operating range of a particular cooling system and optimize its operation. Indices calculated using ion association models also improve the portability of knowledge between waters and systems. The ion association saturation level indices provide a common denominator for comparing results, inhibitor limitations, and failure points where control was lost. They also have served as the driving force for dosage modulation models. And finally, the graphic presentation of complex profiles allows them to be assimilated quickly by water treatment chemists and operational personnel.


1 G. Caplan, Cooling Water Computer Calculations: Do They Compare?, Corrosion '90, Paper 100, National Association of Corrosion Engineers, Houston, Texas, 1990.
2 W.F. Langelier, The Analytical Control of Anti-Corrosion Water Treatment, JAWWA, Vol. 28, No 10, p. 1500-1521, 1936.
3 J.W. Ryznar, A New Index For Determining Amount of Calcium Carbonate Scale Formed By Water, JAWWA, Vol. 36, p. 472, April 1944.
4 P. Puckorius, Get A Better Reading on Scaling Tendency of Cooling Water, Power, p. 79-81, September, 1983.
5 W. Stumm and J.J. Morgan, Aquatic Chemistry, 2nd edition, John Wiley and Sons, New York, New York, pp. ,1981.
6 A.H. Truesdell and B.F. Jones, Wateq - A Computer Program for Calculating Chemical Equilibria of Natural Waters, J. Research, U.S. Geological Survey Volume 2, No. 2, p. 233-248, 1974.
7 W. Chow, J.T. Arson, W.C. Micheletti, Calculations of Cooling Water Systems: Computer Modeling of Recirculating Cooling Water Chemistry, International Water Conference 41rst Annual Meeting, Pittsburgh, Pennsylvania, IWC-84-41.
8 D.A. Johnson, K.E. Fulks, Computerized Water Modeling in the Design and Operation of Industrial Cooling Systems, International Water Conference, 41rst Annual Meeting, Pittsburgh, Pennsylvania, IWC-80-42.
9 R.J. Ferguson, A Kinetic Model for Calcium Carbonate Deposition, Corrosion '84, Paper 120, National Association of Corrosion Engineers, Houston, Texas, 1984.
10 R.J. Ferguson, O. Codina, W. Rule, R. Baebel, Real Time Control of Scale Inhibitor Feed Rate, International Water Conference, 49th Annual Meeting, Pittsburgh, Pennsylvania, IWC-88-57.
11 S. Costa, M.H. Hwang, C.J. McCloskey, The Impact of Computer Models on New Plant Utility Systems, International Water Conference, 51rst Annual Meeting, Pittsburgh, Pennsylvania, IWC-90-46.
12 C.J. Schell, The Use of Computer Modeling in Calguard to Mathematically Simulate Cooling Water Systems and Retrieve Data, International Water Conference, 41rst Annual Meeting, Pittsburgh, Pennsylvania, IWC-80-43.
13 T. Young, The Proper Use of Polymer Technology in Cooling Water Programs, AWT Analyst, Association of Water Technologies, Washington, D.C., January, 1991.
14 R. Ferguson, Computer Aided Proposal Writing, Association of Water Technologies Spring Meeting, 1991.


        Calcium carbonate__ ___ S.L. = ___________
        ____________ _____________ __Ksp CaCO3

        ____________ ____ __________(Ca)(SO4)
        Calcium sulfate________ S.L. = ____________
        ______________ _____________Ksp CaSO4

        Tricalcium phosphate___S.L. = ____________
        _____________ _____________Ksp Ca3(PO4)2

        Amorphous silica ______S.L. = __________________
        _____________ _____________(H2O)2 * Ksp SiO2

        Calcium fluoride_______S.L. = ________
        _____________ _____________Ksp CaF2

        Magnesium hydroxide_ S.L. = ____________
        _____________ _____________Ksp Mg(OH)2

Table 2: Impact of Ion Pairing on LSI


LSI at
Lowest TDS

LSI at
Highest TDS

Impact on LSI

High Chloride
(No Pairing)



0.36 decrease

High Sulfate
(No Pairing)



0.43 decrease

High Chloride
(With Pairing)



0.40 decrease

High Sulfate
(With Pairing)



0.86 decrease

Table 3: Concentration Ratio Limit Comparison

Major Anion

Makeup Sulfate

Makeup Chloride

Concentration Ratio
for Langelier Saturation
Index of 2.5

Concentration Ratio
For Calcite Saturation
Level of 150

Low Sulfate





Sulfate, Chloride Mix















Common Makeup Water Constituents

      Calcium ____120 as CaCO3 ________"M" Alkalinity __110 as CaCO3
      Magnesium __ 23 as CaCO3_________Silica _________36 __as SiO2
      Sodium _ ___180 ___as Na

Table 4: Silica Limits For Three (3) Treatment Schemes

Acid pH 6.0


pH 7.6

High pH 8.9
Lowest Temperature oF







Silica Limit







Saturation Level Limit







Figure 1: Example Ion Pairs Used To
Estimate Free Ion Concentrations

      [Calcium] =______[Ca+II] + [CaSO4] + [CaHCO3+I] + [CaCO3] + [Ca(OH)+I]
      _____________+ [CaHPO4] + [CaPO4-I] + [CaH2PO4+I]

      [Magnesium] = __ [Mg+II] + [MgSO4] + [MgHCO3+I] + [MgCO3] + [Mg(OH)+I]
      _____________+ [MgHPO4] + [MgPO4-I]+[MgH2PO4+I]+[MgF+I]

      [Sodium] = _____[Na+I] + [NaSO4-I] + [Na2SO4] + [NaHCO3] + [NaCO3-I]
      ____________ + [Na2CO3] + [NaCl]+[NaHPO4-I]

      [Potassium] = __ [K+I]+[KSO4-I] + [KHPO4-I] + [KCl]

      [Iron] = _______[Fe+II] + [Fe+III] + [Fe(OH)+I] + [Fe(OH)+II] + [Fe(OH)3-I]
      __________ _+ [FeHPO4+I] + [FeHPO4] + [FeCl+II] + [FeCl2+I] + [FeCl3]
      __________ _+ [FeSO4] + [FeSO4+I] + [FeH2PO4+I] + [Fe(OH)2+I] + [Fe(OH)3]
      ___________+ [Fe(OH)4-I] + [Fe(OH)2] + [FeH2PO4+II]

      [Aluminum] = _[Al+III] + [Al(OH)+II] + [Al(OH)2+I] + [Al(OH)4-I] + [AlF+II] + [AlF2+I]
      __________+ [AlF3] + [AlF4-I] + [AlSO4+I] + [Al(SO4)2-I]

Ion Pairing Reduces LSI (Sulfate Effect Much Greater Than Chloride)

A Langelier Saturation Index of 2.5 coincides with a Calcite Saturation Level of about 150 in low TDS waters.

A Langeleir Index of 2.5 occurs at a lower concentration ratio than a Calcite Saturation Level of 150 in high sulfate waters.

Amorphous silica solubility increases with pH and Temperature.

Control limits may change seasonally in silica limited waters due to the high temperature dependence of amorphous silica solubility.

A silica operating range profile

pH control is critical in Neutral pH range Phosphate Treatment programs .