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Developing Scale Inhibitor Dosage Models

Robert J. Ferguson
French Creek Software, Inc.
Kimberton and Hares Hill Road, Box 684
Kimberton, Pennsylvania 19442 U.S.A.

Presented at WaterTech '92, Houston, Texas
NACE EUROPE '93, Sandefjord, Norway

Published in Industrial Water Treatment Magazine

Feeding the minimum effective inhibitor dosage can reduce operating costs for chemical treatment, minimize treatment chemical discharge to the environment, and in some cases, prevent under­feed of a scale inhibitor. Common sense indicates that the same scale inhibitor dosage is not required for all waters and systems. One size does not fit all. Water treatment companies have capitalized on this general concept since the introduction of first computerized water chemistry evaluation and treatment recommendation systems in the 70's.(1,2,3) Dosage models have been developed by the industry for scale control in applications ranging from long residence time open recirculating cooling tower systems, to the ultra low treatment levels required in very short residence time once through utility surface condenser cooling systems.

This paper discusses the parameters critical to developing an effective dosage modulation model for scale inhibitors from laboratory data, field data, or a combination of both. The paper draws upon the concept of induction time as a basis for the mathematical models used to develop predictive models from actual data. The models are based upon the concept that threshold effect inhibitors do not prevent scale formation, they only delay the inevitable. The models are in agreement with current theories and treat scale inhibitors as agents which extend the induction time before crystal formation and/or growth on existing active sites occurs in the case of calcium carbonate, and as dispersants which control particle size in the case of calcium phosphate.

The models predict the dosage required to inhibit deposition until the treated water has passed through a cooling system. This delay can vary from 3 to 15 seconds in a large volume once through condenser cooling system, to days in open recirculating cooling systems.

Thermodynamic driving forces and system operating conditions are used by the models to describe the kinetics of scale formation, growth, and the impact of inhibitors upon induction time.

Similar models to those discussed have been used successfully to optimize scale inhibitor treatments in once through utility cooling systems and open recirculating cooling systems since the late 70's.

The models described in this paper were developed from a combination of field observations, common sense, and laboratory data. Model development began in the early 1970's. At this time, utilities using man­made impounded lakes as a source for condenser cooling water were beginning to develop condenser scale problems. The lakes were originally filled with water of a low scale potential. The heat load from condenser cooling, and the lack of blowdown, caused the lakes to concentrate with time. Calcium carbonate scale became an economically significant problem after several years of operation. Deration was encountered due to high condenser back pressure in addition to the heat rate penalty associated with increased back pressure due to condenser deposits. Treatment with scale control agents would have cost more than the problem if treatment were implemented using the once through cooling technologies of the time. New technology was needed if the treatment of these systems for scale control was to be economically feasible. Dosage optimization studies were conducted to determine if ultra low dosages, on the close order of 0.001 to 0.2 mg/l active, could effectively prevent calcium carbonate scale.

The initial studies were conducted using well instrumented test heat exchangers. Tests were run in parallel. One exchanger was treated at a high level to assure that scale would be controlled. The parallel exchanger remained untreated. Fouling factors were monitored continuously on the exchangers. Fouling could be measured a day or two prior to the formation of a visible deposit.

The time required for measurable deposit formation was noted. This time was used to determine the minimum time between dosage reductions during the dosage optimization studies. Dosage reductions were made after a minimum of two of these time periods had elapsed.

It was observed that ultra low dosages were effective in preventing scale formation in the short residence time condensers. The cooling water was present in the condensers for less than 10 seconds in all of the systems evaluated. Models were developed based upon the initial studies at seven (7) locations. The once through cooling system data was later expanded to longer residence time open recirculating (cooling tower) systems. Laboratory studies filled data gaps.

During the evaluation of the data it was found that several parameters were critical to dosage: time, temperature, and the degree of supersaturation. System cleanliness was also found to be important. This paper discusses the models and their practical application to cooling water scale control. The method outlined in this paper has been used to develop models of minimum effective inhibitor dosages from laboratory data, field data, and combinations of both. The models provide a natural path for bringing research data into the practical arena of the operating engineer or water chemist.

The use of the models is described in case history format, after a brief description of the impact of individual parameters upon the minimum effective scale inhibitor dosages.

Induction Time: The Key To The Models
Reactions do not occur instantaneously. A time delay occurs once all of the reactants have been added together. They must come together in the reaction media to allow the reaction to happen. The time required before a reaction begins is termed the induction time.

Thermodynamic evaluations of a cooling water scale potential predict what will happen if a water is allowed to sit undisturbed under the same conditions for an infinite period of time. Even simplified indices of scale potential such as the Langelier saturation index can be interpreted in terms of the kinetics of scale formation. For example, calcium carbonate scale formation would not be expected in an operating system when the Langelier saturation index for the system where 0.1 to 0.2 . The driving force for scale formation is too low for scale formation to occur in finite, practical cooling system residence times. Scale would be expected if the same system operated with a Langelier saturation index of 2.8 . The driving force for scale formation in this case is high enough, and induction time short enough, to allow scale formation in even the longest holding time index cooling systems.

Induction time has been modeled for economically important crystals such as sucrose. Models follow a formula similar to equation
_______ Induction Time = _____________________
________________________________k [Saturation Level ­ 1]P­1


Induction Time is the time before crystal formation and growth occurs;
k is a temperature dependent constant;
Saturation Level is the degree of super­saturation;
P is the critical number of molecules in a cluster prior to phase change.

Gill and his associates demonstrated that commercially available scale inhibitors extend the induction time for calcium carbonate scale(4). Their paper points out several critical parameters which impact the induction time prior to crystal growth:

The degree of supersaturation.
The temperature.
The presence of active sites upon which growth can occur.
The inhibitor level.

Gill's study used saturation level as the thermodynamic driving force for scale growth. Saturation level calculations performed using a computerized ion pairing method eliminate most of the assumptions inherent in simplified indices(5). They account for common ion effects which can increase the apparent solubility of a scale forming specie such as calcium carbonate. Driving forces for scale formation calculated using the ion pairing method are transportable between systems because they base their calculations upon free ion concentrations rather than the total analytical values. This is the heart of the ion pairing, or ion association method, which subtracts ion pairs (e.g. CaSO4, CaHCO3­) from the total analytical value to estimate the free ion present and available to react in forming seed crystals, or in driving growth on existing substrates.

The remainder of this paper uses ion association model saturation levels for the driving force for scale formation. Table 1 provides a working definition of the term saturation level for calcium carbonate and tricalcium phosphate.

Critical Parameters
The parameters contributing to equation 1 are included in the basic relationships used for inhibitor dosage modeling. Major data values required include the time period during which scale formation must be prevented, the degree of supersaturation which is the driving force which must be overcome, the temperature at which the inhibitor must function, and the pH of the cooling water. The surface area of active sites also impacts the dosage requirement.

These parameters have the following impacts upon dosage:

Time ­ The time selected is the residence time the inhibited water will be in the cooling system. The inhibitor must prevent scale formation or growth until the water has passed through the system and been discharged. Figure 1 profiles the impact of induction time upon dosage with all other parameters held constant.

Degree of Supersaturation ­ An ion association model saturation level is the driving force for the model outlined in this paper, although other, similar driving forces have been used. Calculation of driving force requires a complete water analysis, and the temperature at which the driving force should be calculated. Figure 2 profiles the impact of saturation level upon dosage, all other parameters being constant.

Temperature ­ Temperature affects the rate constant for the induction time relationship. As in any kinetic formula, the temperature has a great impact upon the collision frequency of the reactants. This temperature effect is independent of the effect of temperature upon saturation level calculations. Figure 3 profiles the impact of temperature upon dosage with other critical parameters held constant.

pH ­ pH affects the saturation level calculations, but it also may affect the dissociation state and stereochemistry of the inhibitors(8). Inhibitor effectiveness can be a function of pH due to its impact upon the charge and shape of an inhibitor molecule. This effect may not always be significant in the pH range of interest (e.g. 6.5 to 9.5 for cooling water).

Active sites ­ It is easier to keep a clean system clean than it is to keep a dirty system from getting dirtier. This rule of thumb may well be related to the number of active sites for growth in a system. When active sites are available, scale forming species can skip the crystal formation stage and proceed directly to crystal growth.

Other factors can impact dosage such as suspended solids in the water. Suspended solids can act as sources of active sites, and can reduce the effective inhibitor concentration in a water by adsorption of the inhibitor. These other factors are not taken into account in the models in this paper. Table 2 summarizes the factors critical to dosage modeling, and their impact upon dosage.

Data Base
The dosage models used as examples in this paper were developed from data collected in field studies(6), laboratory studies, published data, or a combination of these sources.

Examples in this paper include data from sidestream evaluation of the minimum effective dosages in utility surface condensers.(6,7) In these studies, two parallel fouling probes were used to develop estimates of the minimum effective dosages for the phosphonates amino­tris­methylene phosphonic acid (AMP), 1,1­hydroxy ethylidene diphosphonic acid (HEDP), and polyacrylic acid (PAA). One probe was over­treated at a level where no calcium carbonate deposition would be anticipated. The parallel probe was not treated, and the time required for a measurable deposit to form determined. This was deemed the minimum period between dosage adjustments for the test. (Note: A minimum test duration of twice the time required for fouling was allowed to pass between dosage adjustments). Dosages were decreased until failure, as indicated by a measurable deposit formation.

Inhibitor dosages were then decreased to the minimum effective level on the condenser cooling systems to confirm that the dosages did indeed prevent scale. Condenser cleanliness was monitored by heat transfer. This work was done in the late 70's when sub­ppm treatment levels and ultra low dosages were just beginning to be used in utility once through cooling system scale control programs.

A dosage model is only as good as the data from which it is derived. The most generally applicable models include data points over the anticipated ranges for critical parameters. For example, a model developed using data in the temperature range of 30 to 40 ºC might be totally useless in predicting a dosage for a system operating at 70 ºC.

Models should be derived from data over the range of water chemistry anticipated as well as over the range of saturation level anticipated. If a calcium carbonate scale inhibitor model will be used in waters ranging from a calcium level of 40 ppm to over 1000 ppm, this range should be covered from laboratory and/or field sources. The saturation level range anticipated should also be bracketed (e.g. 1.0 to 250 saturation level for calcite).

Although field data is the source of choice, field conditions can rarely be adjusted to cover the temperature, pH, time, and water chemistry ranges desired. The use of static laboratory tests designed to elucidate the variation of dosage with any of the parameters can be used to supplement field data. Field data, although desirable, is not always necessary for the development of a preliminary correlation. As demonstrated in the calcium phosphate deposit control example, dosages predicted by laboratory tests can be directly applicable to field conditions. Each model developed should be compared to field results to assure that a correlation exists between the test data, the model, and actual field results.

Development Of A Model
A modified version of Equation 1 provided the basis for model correlation. Dosage was added as a factor to the equation on the right side to produce Equation 2.

Equation 2 ____
__________________Induction Time = ________________________
______________________________________________k'[Saturation level ­ 1]P­1

The temperature dependent rate constant k' was found to correlate with the Arrhenius relationship (Equation 3).

Equation 3 ______________________k' = A e­Ea/RT

Saturation levels were calculated from water analysis input using a computerized ion association model. The time used for the correlation is the time to failure in laboratory tests, the residence time in a heated state for utility once through cooling systems, and the holding time index in open recirculating cooling systems.

Equation 2 was rearranged to solve for dosage in the first order. Regression analysis was used to estimate the coefficients.

Field Correlation
The test of any model is its applicability to operating systems. Two examples are presented in this paper as an indication of the value of dosage models in suggesting an initial inhibitor treatment level.

Example 1: CaCO3 Scale Control in Utility Once Through Condenser Cooling Systems
In the late 70's the efficacy of ultra­low scale inhibitor dosages was demonstrated in systems serviced by man­made impounded lakes. These system typically started up with a low to moderate hardness water of low to borderline scale potential. The lakes concentrated with time to create a very scaling condition. In many cases, acid cleaning was required to prevent condenser related capability loss, in the absence of treatment.

The minimum effective dosages for these systems ranged from 0.01 to 0.2 ppm active phosphonate, depending upon the water chemistry, temperature, and residence time during which scale deposition or growth had to be prevented. The efficacy of these low level treatments was demonstrated in many of the midwest and south central United States central station power plants where condenser cooling water was supplied by man­made impounded lakes(6,7), and continues to be demonstrated and optimized using on­line real time control(2,3). Real time optimization is an economic necessity in many of these lakes due to the high changes in pH encountered over even a twenty four (24) hour period. pH fluctuations of 1.2 pH units have been reported. As depicted in figure 4, this equates to a ten fold change in dosage requirement in a single day.

Table 3 summarizes the water chemistry, scale potential indices, and dosage recommendation for 100% active HEDP for a single analysis and set of operating parameters for a typical utility once through cooling system as outlined in one of the initial dosage minimization studies(9). The treatment level recommended is comparable to that found effective in the original published study.

It is of interest to note that the model which recommended an accurate treatment level for a short residence time utility once through system also recommends a reasonable treatment level for an open recirculating cooling system with a residence time which can be calculated in days. The model used for this comparison has been found to provide reasonable treatment recommendations for both short and long residence time cooling systems. Figure 5 profiles a typical system.

Example 2: Calcium phosphate Deposition Control
Calcium phosphate inhibitors have been successfully modeled by the method described in the paper. The same basic formula was used for modelling calcium phosphate deposition as was used for the calcium carbonate inhibitors.

Figure 6 indicates the preliminary correlation for a copolymer in common use as a calcium phosphate inhibitor. The model was applied to an open recirculating cooling system water chemistry to determine if the dosages recommended were comparable to those effective in operating systems. The treatment program had undergone extensive dosage optimization to determine the most appropriate treatment level. The recirculating water chemistry, operating parameters, and dosages for the system are outlined in Table 4. The system typically operates at approximately six (6) cycles of concentration.

The model predicted the final dosage within approximately ten (10) percent of the final optimized value. Use of the model would have provided a reasonable initial treatment level for the on­site optimization studies. An initial dosage three times the optimized level was used as a starting point in the actual study.

It is interesting to note that the coefficients calculated for the phosphonate calcium carbonate inhibitors HEDP and AMP were of a comparable order, indicating the same inhibition mechanism. The order for the calcium phosphate scale inhibition by a copolymer is different, indicating a different scale inhibition mechanism.

Laboratory and field dosage optimization data can be converted to a mathematical model using standard statistical methods and a relationship derived from theoretical models for induction time. The models provide a practical method for collating laboratory and field data for a scale inhibitor. The correlations developed can then be used to predict the dosage for cooling systems based upon water chemistry and operating parameters without the necessity for laboratory or in­depth field studies to determine the minimum effective dosage. Dosages predicted by models developed in this manner are typically accurate as long as the system parameters and water chemistry data are within the range of the data used to develop the models. The examples presented in this paper are by necessity limited. The basic models described in this paper have been used successfully in systems ranging from short residence time, low scale potential systems, to high residence time, high scale potential systems for calcium carbonate control. The phosphate models have been used extensively as an integral portion of treatment recommendation systems for multi­functional, alkaline phosphate corrosion and scale control programs.

As with any predictive method, dosage recommendations from such models should be evaluated by an experienced water treatment chemist prior to implementation in an operational cooling system. Predicted dosages should be used as a guideline, not as an ultimate treatment recommendation due to factors which may not be taken into account by the models.


1.  C.J. Schell, "The Use of Computer Modeling in Calguard to Mathematically Simulate Cooling Water Systems and Retrieve Data," paper no. IWC­80­43 (Pittsburgh, PA: International Water Conference, 41rst Annual Meeting, 1980).

2.  R.J. Ferguson, O. Codina, W. Rule, R. Baebel, "Real Time Control of Scale Inhibitor Feed Rate," paper no. IWC­88­57 (Pittsburgh, PA: International Water Conference, 49th Annual Meeting, 1988).

3.  S.R. Payne, B.W. Perrigo, R.M. Post, T.P. Clay, "Application of a Self­calibrating, Microprocessor­driven Metering Device to a Utility Once Through Cooling System," paper no. IWC­90­46 (Pittsburgh, PA: International Water Conference, 51rst Annual Meeting, 1990).

4.  J.S. Gill, C.D. Anderson, R.G. Varsanik, "Mechanism of Scale Inhibition by Phosphonates," paper no. IWC­83­4 (Pittsburgh, PA: International Water Conference, 44th Annual Meeting, 1983).

5.  R.J. Ferguson, "Computerized Ion Association Model Profiles Complete Range of Cooling System Parameters," paper no. IWC­91­47 (Pittsburgh, PA: International Water Conference, 52nd Annual Meeting, 1991).

6.  R.J. Ferguson, "A Kinetic Model for Calcium Carbonate Deposition," CORROSION/84, Paper no. 120, (Houston, TX: National Association of Corrosion Engineers, 1984).

7.  R.J. Ferguson, "Practical Application of Condenser Performance Monitoring to Water Treatment Decision Making," paper no. IWC­81­25 (Pittsburgh, PA: International Water Conference, 42nd Annual Meeting, 1981).

8.  W.M. Hann, J. Natoli, "Acrylic Acid Polymers and Copolymers as Deposit Control Agents in Alkaline Cooling Water Systems," CORROSION/84, Paper no. 315, (Houston, TX: National Association of Corrosion Engineers, 1984).

9.  B.W. Ferguson, R.J. Ferguson, "Sidestream Evaluation of Fouling Factors in a Utility Surface Condenser," Journal of the Cooling Tower Institute,2, (1981):p. 31­39.


Time Dosage increases with residence time.
Degree of Supersaturation 
Dosage increases with saturation level.
Temperature Dosage increases with temperature due to its impact upon reaction rate.

This temperature impact is independent of any impact of temperature upon saturation level.
pH Dosage may be pH dependent due to the impact of pH upon the inhibitor dissociation state and stereochemistry.

This pH impact is independent of any impact of pH upon saturation level.
Suspended solids Dosage requirements may increase as suspended solids increase due to absorbtion of the inhibitor on the solids.
Active sites Dosage requirements increase if active sites for scale growth are present.

It is easier to keep a clean system clean than it is to keep a dirty system from getting dirtier.



Saturation level is the ratio of the Ion Activity
Product to the Solubility Product.

For calcium carbonate:

______SL = _____________

For tricalcium phosphate:

_____SL = _____________

A water will tend to dissolve scale of the
compound if the saturation level is less than 1.0

A water is at equilibrium when the Saturation
Level is 1.0 . It will not tend to form or
dissolve scale.

A water will tend to form scale as the Saturation
Level increases above 1.0 .

Table 3: Utility Once Through Cooling System Example

Lake Water Analysis Deposition Potential Indicators
Cations Saturation Level
Calcium (as CaCO3) 120.0 Calcite (CaCO3) 3.94
Magnesium (as CaCO3) 34.0 Aragonite (CaCO3) 3.84
Sodium (as Na) 14.00 Silica (SiO2) 0.03
Potassium (as K) 0.00 Calcium phosphate (Ca3(PO4)2) 0.00
Iron (as Fe) 0.10 Anhydrite (CaSO4) 0.00
Ammonia (as NH3) 0.00 Gypsum (CaSO4 * 2H2O) 0.00
Aluminum (as Al) 0.00 Fluorite (CaF2) 0.00
Boron (as B) 0.00 Brucite (Mg(OH)2) 0.01
Anions Simple Indices
Chloride (as Cl) 35.0 Langelier 1.07
Sulfate (as SO4) 13.0 Ryznar 6.27
"M" Alkalinity (as CaCO3) 120.0 Practical 6.64
"P" Alkalinity (as CaCO3) 0.0 Larson-Skold 0.39
Silica (as SiO2) 7.0 Treatment Recommendation
Phosphate (as PO4) 0.0 10% HEDP (mg/L) 0.20
Fluoride (as F) 0.0 Parameters
Nitrate (as NO3) 0.0 pH 8.40
Other Temperature (oC) 20.0
Calculated TDS 254 Residence Time (Seconds) 5.60

Table 4: Calcium Phosphate Inhibitor Dosage Optimization Example

Lake Water Analysis Deposition Potential Indicators
Cations Saturation Level
Calcium (as CaCO3) 1339. Calcite (CaCO3) 38.77
Magnesium (as CaCO3) 496. Aragonite (CaCO3) 32.90
Sodium (as Na) 1240. Silica (SiO2) 0.39
Potassium (as K) 0.00 Calcium phosphate (Ca3(PO4)2) 1,074.
Iron (as Fe) 0.00 Anhydrite (CaSO4) 1.33
Ammonia (as NH3) 0.00 Gypsum (CaSO4 * 2H2O) 1.67
Aluminum (as Al) 0.00 Fluorite (CaF2) 0.00
Boron (as B) 0.00 Brucite (Mg(OH)2) 0.01
Anions Simple Indices
Chloride (as Cl) 620. Langelier 1.99
Sulfate (as SO4) 3,384. Ryznar 4.41
Bicarbonate (as HCO3) 294.1 Practical 4.20
Carbonate (as CO3) 36.2 Larson-Skold 0.39
Silica (as SiO2) 62.0 Treatment Recommendation
Phosphate (as PO4) 6.20 100% Active Copolymer (mg/L) 9.53
Fluoride (as F) 0.0 Parameters
Nitrate (as NO3) 0.0 pH 8.40
Other Temperature (oC) 20.0
Calculated TDS 609 Residence Time (Seconds) 5.60

Delaying The Inevitable: Induction time increases as scale inhibitor dosage increases

The Dosage required to prevent scale formation increases as Saturation Level increases.

The Inhibitor Dosage required to prevent scale increases as Temperature increases.

Dosage increases as a function of Calcite Saturation Level in this example.

Dosage increases with concentration ratio in this example.