analyst 2004 winter ph values

8/2/2019 Analyst 2004 Winter pH Values

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Analyst Winter 2004 - The pH Values for Cooling Water Systems Page 1 of 9

SPRING 2004 | Volume xi | Number 2

The pH Values for Cooling WaterSystems

By Dan Vanderpool, Laurel Functional Chemicals

AbstractThere are several important pH values for cooling water systems. These pHsare used to understand the potential for scale formation. They are also used toassess a pilot cooling towers ability to a model an actual cooling tower andto design research and laboratory evaluations. This article explains each of

these pHs and the calculations used to derive them.Key Words: Ion Equilibrium Model, Langelier Saturation Index, RyznarStability Index, Practical Scaling Index, Regression Method, AtmosphericEquilibrium, Temperature effect on pH, pHs.

IntroductionpH has a pivotal role in the chemistry of water. It influences the drivingforces for scale and corrosion. Yet, a measured pH value becomes meaningfulonly when it is put into context with the other important pH values that arecharacteristic of the water. These characteristic pH values are derived frombasic ionic equilibria and from empirically established relationships. The

commonly used pH relationships in Table 1 are important benchmarks forcooling water. The example water shown in Table 2 will be used to computethese pH values.

3 (1)

3 (2)

(3)

3 (4)

(5)

(6)

3) (7)

2 (8)

](9)

] 2 (10)

3 (11)

Table 1: Characteristic pH Relationships for Cooling Water

Kunz: pH = 1.6 Log (Alk as mg/L CaCO ) + 4.4

Caplan: pH = 1.8 Log (Alk as mg/L CaCO ) + 4.0

LSI = pH pHs

LSI = approximate Log(CaCO Saturation)

RSI = 2 pHs pH

PSI = 2 pHs pHe

pHe = 1.46 Log (Alk as mg/L CaCO + 4.54

Langelier: pHs = -Log [H] = Log (Ksp/K ) + p[Ca] + p(Alk)

Larson & Buswell: pHs = Log (Ksp/K2) + p[Ca] + p(Alk) + [2.5(I)1/2]/[1+5.3(I)1/2+ 5.5I

Alk at pHs = -[H] + Kw/[H + [H](Ksp/K [Ca]) + 2 Ksp/[Ca]

Atmospheric Equilibrium pH = 0.92 Log (Alk as mg/L CaCO ) + 6.6

http://www.awt.org/members/publications/analyst/2004/spring/the_ph_values.htm 20/03/2005


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3

Alkalinity as CaCO3

4=

Temperature 77 oF

pHs @ 77 oF 6.59

pHe 8.60

) pH 9.2

Table 2: Composition of Example Water Used for Calculation Comparisons

Calcium as CaCO 375 mg/L

600 mg/L

Chloride as Cl 266 mg/L

Sulfate as SO 14 mg/L

Conductivity 1455 mho

Atmospheric Equilibrium (Atm Equil

The Typical Cooling Tower pHThe operating pH of a cooling tower unfortunately cannot be accuratelyforecasted from equilibrium equations. The reason is that typical coolingwater has a lower pH than expected due to excess carbon dioxide arising fromneutralization of alkalinity and biological respiration. Forecasting theoperating pH requires another approach. In the mid-1970s, R. D. Kunz tooksamples from operating cooling tower waters to determine an empirical

relationship for forecasting the operating pH.1 Caplan updated this equation

by averaging nine other empirical relationships.

2

Equations 1 and 2 show thetwo relationships.Equations 1 and 2 are used for estimating the alkalinity required achieving atarget pH in cycled-up cooling water, and for calculating the acid needed toneutralize excess alkalinity in the make-up. They are also used for comparingthe operation of a cooling-tower with typical towers, and in evaluating pilotcooling-towers as mimics of real towers.

pHs and Scaling IndexesThe pHs is the pH at which water has a calcium carbonate saturation of oneand it is commonly used to quantify calcium carbonate scaling tendency.

Langelier defined the scaling tendency for calcium carbonate, the LSI, as thedifference between the operating pH and pHs, as shown by equation 3. Thisvalue is approximately equal to the Log of CaCO3 saturation as shown by

equation 4.An important limitation of the LSI is the non-consideration of the chemical

complexes of calcium, and approximation of the carbonate/bicarbonate ratioinherent in the pH difference. The LSI begins to significantly deviate from

the true CaCO3 saturation above pH 8.0; and above pH 10, where the CaOH+

complex becomes important. The LSI becomes meaningless with regard toCaCO3 saturation. Table 3 shows the deviation of the LSI saturation from the

true saturation.Another measure of scaling tendency based on pHs was developed by

Ryznar.3 He recognized that above pH 10 the LSI would falsely predictscaling where, in actuality, the water was corrosive, i.e., the water would



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saturation for calcium carbonate according to the equation CaCO3 solid + H+

-= Ca++ + HCO3 , namely, ([Ca][HCO3]/[H])/(Ksp/K2) = 1; where, Ksp =

[Ca][CO3] for calcite, and K2 = [H][CO3]/[HCO3] for the second dissociation

constant of carbonic acid.Langelier derived his formula for pHs by rearranging this equation to give

[H] = (K2/Ksp)[Ca][HCO3]. And thus, pHs = -Log [H] = Log (Ksp/K2) + p[Ca] + p[HCO3]. All the terms on the right side are known except [HCO3], so

Langelier used the definition of Alkalinity to calculate [HCO3] as follows:

Alk = -[H] + Kw/[H] + [HCO3] + 2K2[HCO3]/[H], where, [OH] = Kw/[H];

and [CO3] = K2[HCO3]/[H]. Upon rearrangement one obtains [HCO3] = (Alk

+ [H] Kw/[H])/(1 + 2K2/[H]); or, p[HCO3] = p(Alk + [H] Kw/[H]) - p(1 +

2K2/[H]). Since Alk >> [H] and Kw/[H]; and 1 > 2K2/[H], where K2 is

approximately 10-10.3 and [H] is approximately 10-7, then p[HCO3] =

approximately p(Alk). Thus, pHs is given by Langelier as pHs = -Log [H] =Log (Ksp/K2) + p[Ca] + p(Alk), as shown by equation 8.

Langelier therefore used an approximate value of [HCO3]; moreover, hedid not correct the values of Ksp, K2, and [H] for dissolved salts, althoughthe values of Ksp and K2 were corrected for temperature.

To improve the accuracy for cooling water, Larson & Buswell added a newterm to approximately correct for dissolved salts to give: pHs = Log (Ksp/K2)

+ p[Ca] + p(Alk) + [2.5(I)1/2 ]/[1+ 5.3(I)1/2 + 5.5I], as shown by equation 9.In contrast to these approximate approaches a precise way to calculate pHs

is the regression method, which rigorously calculates the bicarbonateconcentration and incorporates dissolved salts corrections. It starts by

combining the equation for unit saturation and the definition of alkalinity togive an equation for the unknown [H] (where [Ca] is equal to the calciumhardness): Alk = -[H] + Kw/[H] + [H](Ksp/K2[Ca]) + 2Ksp/[Ca], as shown

by equation 10. The solution to this equation using the Newton-Ralphsonmethod is given in Appendix 1. Then, [H] is corrected by its activitycoefficient to yield pHs, i.e., pHs = -Log ([H]fh).

A Comparison of these three ways of calculating pHs is shown in Table 5

for the example water at 77 oF.

Comment

6.29 3]

6.49 3]

6.59 3]

3 3

.

Table 5: Comparison of Three Methods of Calculating pHs

Method pHs

Langelier original No ionic strength; approximate [HCO

Larson & Buswell Uses approximate ionic strength; approximate [HCO

Regression Uses exact ionic strength; exact [HCO

Notice that pHs varies by up to 0.3 units between the calculations. This translates to a two

fold variation in terms of CaCO saturation according to the relation (CaCO saturation) =

approximately 10LSI

The pH in the Heat ExchangerpH decreases when temperature is raised in a confined space where CO2



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cannot escape. Langelier believed it was important to account for this effect.4

The true scaling tendency in the heat exchanger, where scaling is usuallymost problematic, requires the use of this lower pH. The conversion of thepH of the basin water to the pH in the heat exchanger requires two separatecalculations.

Here is how Langelier approached the two calculations. First, the dissolved

carbon dioxide concentration is determined from the conditions at themeasured pH using the Langelier definition of alkalinity: Alk = -[H] + Kw/

[H] + K1[CO2*]/[H] + 2K2K1[CO2*]/[H]2, where, [OH ] = Kw/[H]; [HCO3]

= K1 [CO2*]/[H]; [CO3] = K2[HCO3]/[H]; and, the equilibrium constants are

for the temperature of pH measurement. Since [H] is known from themeasured pH, this equation is now a function of [CO2*]. [CO2*] can be

found by the regression equations given in Appendix 2. The results yield thetotal carbon dioxide at the measured pH as: Tco2 = [CO2*] + [HCO3] +

[CO3].

Since Tco2 stays the same while the water is inside the heat exchanger, it isused along with the alkalinity equation, to calculate the redistribution ofcarbonate species and pH at the second temperature. The calculation uses theequilibrium constants for the second temperature, Kw, K1, and K2. See the

spreadsheet on the AWT websitewww.awt.org\members\publications\analyst\2004

\spring\Langelier_SatCalc1.htm for a working layout of the calculations.Table 6 shows the effect of temperature adjustment for the example water in

Table 2, with measured pH of 8.0 at 77 oF, and heat exchanger temperature of

112 oF.

Value

Corrected for Temp

pH oF) oF)

oF 6.44

oF 1.56 1.49

Approx CaCO3

10

36 31

3

Table 6: Effect of Temperature Correction of pH on Calcium Carbonate Saturation

Uncorrected for Temp

8.0 (measured at 77 7.93 (at 112

pHs for 112 6.44

LSI = pH pHs for 112

Saturation =

LSI

In this example, the temperature adjustment of the pH has a relatively small effect of only0.07 LSI units, which arises from the lower pH in the heat exchanger; this results in a 15

% difference in the CaCO saturation values.

The Atmospheric Equilibrium pHThe atmospheric equilibrium pH is the pH achievable when pH-control is notpracticed and provided that no calcium carbonate precipitation occurs.Knowledge of the atmospheric equilibrium pH will define how closely acooling system is operating to its natural pH limit and may help to obtain

more cycles from a given make-up water.For example, typical municipal potable water used as makeup has a higher

pH than expected because it contains too little carbon dioxide. The pH can belowered to the atmospheric equilibrium pH by adding a small amount of



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carbon dioxide, potentially allowing greater cycling before calcium carbonate scale becomes limiting. The atmospheric equilibrium pH is calculated from Henrys Constant, Kg, and the partial pressure of carbon dioxide in the atmosphere, Pco2, which is typically between 10(-3.39)and 10(-3.61) atmospheres. In order to calculate the atmospheric equilibrium pH, Langelier used the following equilibria to define the species: CO2 gas = CO2 aq, where [CO2 aq] = Kg Pco2;CO2 aq + H2O = H2CO3, where Kh = [H2CO3]/[CO2 aq];CO2* = [CO2 aq] + [H2CO3], so[CO2*] = [CO2 aq](1 + Kh) = approximately [CO2 aq] since Kh =approximately 10(-2.8), also [CO2*] is used to define the first dissociation constant for carbonic acid, thus [HCO3-] = K1[CO2*]/[H].

The Langelier definition of alkalinity then becomes on expansion: Alk = -

H + Kw/[H] + K1(KgPco2)/[H] + 2K2K1(KgPco2)/[H]2. The Langelier Alk

equation can be solved for [H] by the regression method as in Appendix 3.See the spreadsheet on the AWT websitewww.awt.org\members\publications\analyst\2004\spring\AtmEquil_pH_Langelier.htm for working spreadsheet.

The calculation can be made more precise by adding the chemical speciesfor hardness and its complexes with carbonates, hydroxide and sulfate. Thisrequires adding the mass balance equations for hardness ([M] = sum ofhardness ions) and Total Sulfate: Alk = -[H] + [OH] + [HCO3] + 2[CO3] +

[MHCO3] + 2[MCO3] + [MOH]; Total Hardness = [M] + [MHCO3] +

[MCO3] + [MOH] + [MSO4]; and Total SO4 = [SO4] +[MSO4]. The solutionof this expanded model can be achieved with the regression method forsimultaneous equations as demonstrated in the spreadsheet,AtmEquil_pH_Expanded.

Alternatively, a much easier way to calculate of the atmospheric

equilibrium pH is the alkalinity/pH relation equation 11.5

How do these calculations of atmospheric equilibrium pH compare? Table7 shows the results for the example water.

2* -3.5 -3.61 -3.39

9.14 9.23 9.05

9.19 9.27 9.10

9.16 9.16 9.16

*Pco2

3)

2

oF.

Table 7: Three Methods of Calculating Atmospheric Equilibrium pH

Calculated Atmospheric Equilibrium pH:

Log Pco

Langelier model

Expanded model

Quick Formula**

is the carbon dioxide partial pressure in units of atmospheres.

**Atmospheric Equilibrium pH = 0.92 Log (Alk as mg/L CaCO + 6.6,

for Pco = 10-3.5; 77

The simple Langelier model and the Quick Formula give the equilibrium pH within 0.05of a pH unit of the more rigorous Expanded model. In contrast, variation in the normal



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atmospheric carbon dioxide partial pressure of between 10-3.39 and 10-3.61 atmospherescauses greater uncertainty of 0.17 pH units.

ConclusionsThe various pH values used to characterize the calcium carbonate scale

potential and tower operation have been described, along with their methodof calculation.

The pH of unit saturation, pHs, is important as it is used in several scalingindexes; the most widely used being the LSI. The LSI is important because itis approximately equal to the logarithm of the true calcium carbonatesaturation. Depending on the level of approximation in the calculation of thepHs, pHs can vary by several tenths of a pH unit, or up to two-fold variationin terms of CaCO3 saturation values.

The influence of temperature on pH is described: the pH decreases as thetemperature is raised (in a confined space where CO2 cannot escape). The

temperature affect typically changes the LSI by about a tenth of a pH unit, orapproximately 15 % variation in terms of CaCO3 saturation values.

Three ways of calculating the atmospheric equilibrium pH are compared.The atmospheric equilibrium pH is the pH attainable when acid is not fed tothe cooling water and provided no CaCO3 precipitation occurs. The

rudimentary Langelier model and the Quick Formula give values ofatmospheric equilibrium pH within a few hundredth pH units of the morerigorous Expanded model. In contrast, the choice of atmospheric carbondioxide partial pressure causes a greater uncertainty of almost 0.2 pH units.

The various pH values can be calculated using the regression formulas inthis article with common spreadsheet computer programs like Microsoft

Excel, Lotus 123, etc. In setting up even the simplest ionic equilibriummodel, one sees the paradoxical nature of dissolved carbon dioxide; namely,its concentration can affect the pH but not the alkalinity.

Appendixes

Appendix 1. Equations used in the Regression Method for pHs.(See Langelier_SatCalc for a working example).

The regression equation for Alkalinity, Alk = -[H] + Kw/[H] + [H](Ksp/K2

[Ca]) + 2Ksp/[Ca], is solved using an initial guess value for [H]. Thedifference between the calculated value of Alk and the true Alk is used tofind an improvement to the initial guess of [H]. The improvement is foundwith the formulas below:

Alk = (Alktrue Alkcalc)

[H] = [H guess] Alk/(-[H guess] Kw/[H guess] + [H guess] Ksp/(K2[Ca])

+ 2 Ksp/[Ca])[H] better = [H] guess + [H]

Alk is repeatedly calculated with [H] + [H] so that Alk is very small.Generally, 4 repetitions give the correct value of [H] so that Alk < 10-20.pHs is calculated as Log ([H]fh), where fh is the activity coefficient for [H].



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Appendix 2. Regression formulas for solving for [CO2*] concentration.(See Langelier_SatCalc for a working example). When the pH is given, the unknown in the alkalinity equation, Alk = -[H] + Kw/[H] + K1[CO2*]/[H] + 2 K2K1[CO2]/[H]

2, is the dissolved carbon dioxide since [H] = 10

-pH

/fh. These are the equations used in the Newton-

Ralphson method: Alk = Alk true Alk calc

Alk/(-[H] + Kw/[H] + K1[CO2*]/[H] + 2K2K1[CO2*]/[H]2) [CO2*] better = [CO2*] guess + CO2*

CO2*=[CO2*]

Appendix 3. Regression equations to calculate the Langelier AtmosphericEquilibrium pH. (See AtmEquil_pH_Langelier for a working example).

Solve the Alk equation for atmospheric equilibrium, Alk = -H + Kw/[H] + K1

(KgPco2)/[H] + 2K2K1(KgPco2)/[H]2, by making a guess value of [H]6,7

then calculate improvements with the formula: [H] = [H]( Alk)/(-[H]

Kw/[H] - K1KgPco2/[H] - 4K2K1KgPco2/[H]2). The correct concentration of

[H] is converted into activity to give pH, i.e., Atmospheric Equilibrium pH =-Log ([H]fh).

Web links to spreadsheets for doing the calculations described in thisarticle:True CaCO3 Saturation

(See Expanded_SatCalc2.htm)

pHs Regression methodDissolved Carbon Dioxide, [CO2*] Temperature Correction of pH

(See Langelier_SatCalc1.htm)

Atmospheric Equilibrium pH,Langelier Model(See AtmEquil_pH_Langelier.htm)

Atmospheric Equilibrium pH, Expanded Model(See AtmEquil_pH_Expanded.htm)

References:

1. R.D. Kunz, A.F. Yen, and T.C. Hess, Cooling Water Calculations,Chemical Engineering, (August 1977) pp 62-71.

2. Gary Caplan, Cooling Water Computer Calculations: Do TheyCompare? Corrosion /90, NACE International Corrosion Forum, LasVegas, NV (April 1990)

3. John W. Ryznar, A New Index for Determining Amount of CalciumCarbonate Scale Formed by a Water, J Amer Water Works Assoc(April 1977, vol 36) pp 472-486

4. W.F. Langelier, Effect of Temperature on the pH of Natural Waters,



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J Amer Water Works Association, vol 38 (1946) pp 179-1855. Dan Vanderpool, Hidden Assumptions in Saturation Calculations,

The Analyst, vol IX, No. 3, 2002, p. 25.6. H. Margenau and R. M. Murphy, Mathematics of Physics and

Chemistry, (Van Nostrand Co., NY 1943) p. 4767. The Atmospheric Equilibrium pH equation is a quadratic with respect

to the variable [H]; there are mathematically two correct solutions. Toavoid obtaining the inappropriate solution, the initial guess of [H] mustbe higher than the correct value, while not being too high. The authoruses the formula pH = 0.9log(Alk as mg/L CaCO3) + 7.2. [H] isapproximately equal to 10-pH. If a low initial guess is used, theconvergence finds a physically meaningless solution (equal to thealkalinity.) If too high a guess is used, the first iteration over correctsand jumps to the convergence of the physically meaningless solution.

About the Author:Dr. Dan Vanderpool is president of Laurel Functional Chemicals, 2925 32ndStreet, Northport, AL 35476, Phone:(205)-339-5718.

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