ph values for cooling water systems

8/13/2019 pH Values for Cooling Water Systems

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The pH Values for Cooling WaterSystemsBy Dan Vanderpool, Laurel Functional Chemicals

AbstractThere are several important pH values for cooling water systems. These pHs are used tounderstand the potential for scale formation. They are also used to assess a pilot cooling

towers ability to a model an actual cooling tower and to design research and laboratory

evaluations. This article explains each of these pHs and the calculations used to derive

them.Key Words: Ion !uilibrium "odel# $angelier %aturation Index# &y'nar %tability Index#

(ractical %caling Index# &egression "ethod# )tmospheric !uilibrium# Temperature

effect on pH# pHs.

IntroductionpH has a pivotal role in the chemistry of water. It influences the driving forces for scaleand corrosion. *et# a measured pH value becomes meaningful only when it is put into

context with the other important pH values that are characteristic of the water. These

characteristic pH values are derived from basic ionic e!uilibria and from empirically

established relationships. The commonly used pH relationships in Table + are importantbenchmar,s for cooling water. The example water shown in Table - will be used to

compute these pH values.

Table 1: Characteristic pH Relationships for Cooling Water

Kun': pH +./ $og 0)l, as mg1$ 2a2345 6 7.7 0+5

2aplan: pH +.8 $og 0)l, as mg1$ 2a2345 6 7.9 0-5

$%I pH pHs 045

$%I approximate $og02a234%aturation5 075

&%I - pHs pH 0;5

(%I - pHs pHe 0/5

pHe +.7/ $og 0)l, as mg1$ 2a2345 6 7.;7 0H? $og 0Ksp1K-5 6 p>2a? 6 p0)l,5 085

$arson @ Auswell: pHs $og 0Ksp1K-5 6 p>2a? 6 p0)l,5 6 >-.;0I5+1-?1>+6;.40I5+1-

6 ;.;I?0B5

)l, at pHs =>H? 6 Kw1>H? 6 >H?0Ksp1K->2a?5 6 - Ksp1>2a? 0+95

)tmospheric !uilibrium pH 9.B- $og 0)l, as mg1$ 2a2345 6 /./ 0++5


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Calculation of pHsThere are several formulas used to calculate pHs. They differ by the approximations used

in the calculations. They share the common definition of al,alinity# defined by $angelieras consisting only of hydrogen# hydroxide# bicarbonate and carbonate ions. 0ionic charges

are not shown for clarity5:)l, =>H? 6 >3H?6 >H234?6 ->234?. They also share the definition of unit saturation forcalcium carbonate according to the e!uation 2a234 solid6 H

6 2a666 H234=# namely#

0>2a?>H234?1>H?510Ksp1K-5 + where# Ksp >2a?>234? for calcite# and K- >H?>234?1

>H234? for the second dissociation constant of carbonic acid. $angelier derived his formula for pHs by rearranging this e!uation to give >H?

0K-1Ksp5>2a?>H234?. )nd thus# pHs =$og >H? $og 0Ksp1K-5 6 p>2a? 6 p>H234?. )ll

the terms on the right side are ,nown except >H23 4?# so $angelier used the definition of)l,alinity to calculate >H234? as follows: )l, =>H? 6 Kw1>H? 6 >H234? 6 -K->H234?1

>H?# where# >3H? Kw1>H? and >234? K->H234?1>H?. pon rearrangement one obtains

>H234? 0)l, 6 >H? Kw1>H?510+ 6 -K-1>H?5 or# p>H234? p0)l, 6 >H? Kw1>H?5 = p0+

6 -K-1>H?5. %ince )l, LL >H? and Kw1>H? and + L -K-1>H?# where K-is approximately+9=+9.4and >H? is approximately +9=H234? approximately p0)l,5. Thus# pHs is

given by $angelier as pHs =$og >H? $og 0Ksp1K-5 6 p>2a? 6 p0)l,5# as shown by

e!uation 8. $angelier therefore used an approximate value of >H234? moreover# he did not

correct the values of Ksp# K-# and >H? for dissolved salts# although the values of Ksp and

K- were corrected for temperature. To improve the accuracy for cooling water# $arson @ Auswell added a new term to

approximately correct for dissolved salts to give: pHs $og 0Ksp1K-5 6 p>2a? 6 p0)l,5 6

>-.;0I5+1-?1>+6 ;.40I5+1-6 ;.;I?# as shown by e!uation B.

In contrast to these approximate approaches a precise way to calculate pHs is the

regression method# which rigorously calculates the bicarbonate concentration andincorporates dissolved salts corrections. It starts by combining the e!uation for unit

saturation and the definition of al,alinity to give an e!uation for the un,nown >H? 0where>2a? is e!ual to the calcium hardness5: )l, =>H? 6 Kw1>H? 6 >H?0Ksp1K->2a?5 6 -Ksp1

>2a?# as shown by e!uation +9. The solution to this e!uation using the Jewton=&alphson

method is given in )ppendix +. Then# >H? is corrected by its activity coefficient to yieldpHs# i.e.# pHs =$og 0>H?fh5.

) 2omparison of these three ways of calculating pHs is shown in Table ; for the

example water at H234?

$arson @ Auswell /.7B ses approximate ionic strength approximate >H234?

&egression /.;B ses exact ionic strength exact >H234?


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Jotice that pHs varies by up to 9.4 units between the calculations. This translates to a two=fold variation in terms of 2a234saturation according to the relation 02a234saturation5

approximately +9$%I.

The pH in the Heat ExchangerpH decreases when temperature is raised in a confined space where 23-cannot escape.

$angelier believed it was important to account for this effect.7The true scaling tendency

in the heat exchanger# where scaling is usually most problematic# re!uires the use of this

lower pH. The conversion of the pH of the basin water to the pH in the heat exchangerre!uires two separate calculations.

Here is how $angelier approached the two calculations. Dirst# the dissolved carbon

dioxide concentration is determined from the conditions at the measured pH using the$angelier definition of al,alinity: )l, =>H? 6 Kw1>H? 6 K+>23-M?1>H? 6 -K-K+>23-M?1

>H?-# where# >3H ? Kw1>H? >H234? K+ >23-M?1>H? >234? K->H234?1>H? and# the

e!uilibrium constants are for the temperature of pH measurement. %ince >H? is ,nownfrom the measured pH# this e!uation is now a function of >23-M?. >23-M? can be found by

the regression e!uations given in )ppendix -. The results yield the total carbon dioxide atthe measured pH as: Tco- >23-M? 6 >H234? 6 >234?.

%ince Tco-stays the same while the water is inside the heat exchanger# it is used alongwith the al,alinity e!uation# to calculate the redistribution of carbonate species and pH at

the second temperature. The calculation uses the e!uilibrium constants for the second

temperature# Kw# K+# and K-. %ee the spreadsheet on the )WT websitewww.awt.orgNmembersNpublicationsNanalystN-997NspringN$angelierO%at2alc+.htm for a

wor,ing layout of the calculations. Table / shows the effect of temperature adPustment

for the example water in Table -# with measured pH of 8.9 at


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Gnatural pH limit and may help to obtain more cycles from a given ma,e=up water.

Dor example# typical municipal potable water used as ma,eup has a higher pH than

expected because it contains too little carbon dioxide. The pH can be lowered to theatmospheric e!uilibrium pH by adding a small amount of carbon dioxide# potentially

allowing greater cycling before calcium carbonate scale becomes limiting.

The atmospheric e!uilibrium pH is calculated from Henrys 2onstant# Kg# and the partialpressure of carbon dioxide in the atmosphere# (co-# which is typically between +90=4.4B5and

+90=4./+5atmospheres. In order to calculate the atmospheric e!uilibrium pH# $angelier used

the following e!uilibria to define the species:

23-gas 23-a!# where >23-a!? Kg(co-23-a! 6 H-3 H-234# where Kh >H-234?1>23-a!?

23-M >23-a!? 6 >H-234?# so

>23-M? >23-a!?0+ 6 Kh5 approximately >23-a!? since Kh approximately +90=-.85#

also >23-M? is used to define the first dissociation constant for carbonic acid# thus

>H234=? K+>23-M?1>H?.

The $angelier definition of al,alinity then becomes on expansion: )l, =H 6 Kw1>H? 6K+0Kg(co-51>H? 6 -K-K+0Kg(co-51>H?-. The $angelier )l, e!uation can be solved for >H?

by the regression method as in )ppendix 4. %ee the spreadsheet on the )WT website

www.awt.orgNmembersNpublicationsNanalystN-997NspringN )tm!uilOpHO$angelier.htm

for wor,ing spreadsheet.The calculation can be made more precise by adding the chemical species for hardness

and its complexes with carbonates# hydroxide and sulfate. This re!uires adding the mass

balance e!uations for hardness 0>"? sum of hardness ions5 and Total %ulfate: )l, =>H? 6 >3H? 6 >H234? 6 ->234? 6 >"H234? 6 ->"234? 6 >"3H? Total Hardness >"?

6 >"H234? 6 >"234? 6 >"3H? 6 >"%37? and Total %37 >%37? 6>"%37?. The

solution of this expanded model can be achieved with the regression method for

simultaneous e!uations as demonstrated in the spreadsheet# )tm!uilOpHOxpanded. )lternatively# a much easier way to calculate of the atmospheric e!uilibrium pH is the

al,alinity1pH relation e!uation ++.;

How do these calculations of atmospheric e!uilibrium pH compare Table < shows theresults for the example water.

Table $: Three ,ethods of Calculating .tmospheric E/uilibrium pH

Calculated .tmospheric E/uilibrium pH:

$og (co-M =4.; =4./+ =4.4B

$angelier model B.+7 B.-4 B.9;

xpanded model B.+B B.-< B.+9

Ruic, DormulaMM B.+/ B.+/ B.+/

M(co-is the carbon dioxide partial pressure in units of atmospheres.MM)tmospheric !uilibrium pH 9.B- $og 0)l, as mg1$ 2a2345 6 /./#

for (co- +9=4.;


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The simple $angelier model and the Ruic, Dormula give the e!uilibrium pH within 9.9; of

a pH unit of the more rigorous xpanded model. In contrast# variation in the normal

atmospheric carbon dioxide partial pressure of between +9=4.4Band +9=4./+atmospheres

causes greater uncertainty of 9.+< pH units.

ConclusionsThe various pH values used to characteri'e the calcium carbonate scale potential and

tower operation have been described# along with their method of calculation.The pH of unit saturation# pHs# is important as it is used in several scaling indexes the

most widely used being the $%I. The $%I is important because it is approximately e!ual

to the logarithm of the true calcium carbonate saturation. Fepending on the level ofapproximation in the calculation of the pHs# pHs can vary by several tenths of a pH unit#

or up to two=fold variation in terms of 2a234saturation values.

The influence of temperature on pH is described: the pH decreases as the temperature is

raised 0in a confined space where 23-cannot escape5. The temperature affect typicallychanges the $%I by about a tenth of a pH unit# or approximately +; Q variation in terms

of 2a234saturation values. Three ways of calculating the atmospheric e!uilibrium pH are compared. The

atmospheric e!uilibrium pH is the pH attainable when acid is not fed to the cooling water

and provided no 2a234precipitation occurs. The rudimentary $angelier model and the

GRuic, Dormula give values of atmospheric e!uilibrium pH within a few hundredth pHunits of the more rigorous xpanded model. In contrast# the choice of atmospheric carbon

dioxide partial pressure causes a greater uncertainty of almost 9.- pH units.

The various pH values can be calculated using the regression formulas in this articlewith common spreadsheet computer programs li,e "icrosoft xcelS# $otus +-4S# etc.

In setting up even the simplest ionic e!uilibrium model# one sees the paradoxical natureof dissolved carbon dioxide namely# its concentration can affect the pH but not theal,alinity.

Appendixes

.ppendix 1%!uations used in the &egression "ethod for pHs.0%ee $angelierO%at2alcfor a wor,ing example5.

The regression e!uation for )l,alinity# )l, =>H? 6 Kw1>H? 6 >H?0Ksp1K->2a?5 6 -Ksp1

>2a?# is solved using an initial guess value for >H?. The difference between the calculated

value of )l, and the true )l, is used to find an improvement to the initial guess of >H?.The improvement is found with the formulas below:

)l, 0)l, true )l, calc5

>H? >H guess? )l,10=>H guess? Kw1>H guess? 6 >H guess? Ksp10K->2a?5 6 - Ksp1>2a?5

>H? better >H? guess6 >H?
http://www.awt.org/members/publications/analyst/2004/spring/Langelier%20SatCalc1.htmhttp://www.awt.org/members/publications/analyst/2004/spring/Langelier%20SatCalc1.htm


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)l, is repeatedly calculated with >H? 6 >H? so that )l, is very small. enerally# 7

repetitions give the correct value of >H? so that )l, U +9=-9. pHs is calculated as $og

0>H?fh5# where fhis the activity coefficient for >H?.

.ppendix 2%&egression formulas for solving for >23-M? concentration.

0%ee $angelierO%at2alcfor a wor,ing example5.

When the pH is given# the un,nown in the al,alinity e!uation# )l, =>H? 6 Kw1>H? 6

K+>23-M?1>H? 6 - K-K+>23-?1>H?-# is the dissolved carbon dioxide since >H? +9 =pH1fh.

These are the e!uations used in the Jewton=&alphson method:

)l, )l, true )l, calc

23-M>23-M? )l,10=>H? 6 Kw1>H? 6 K+>23-M?1>H? 6 -K-K+>23-M?1>H?-5

>23-M? better >23-M? guess 6 23-M

.ppendix 3%&egression e!uations to calculate the $angelier )tmospheric !uilibrium

pH. 0%ee )tm!uilOpHO$angelierfor a wor,ing example5.

%olve the )l, e!uation for atmospheric e!uilibrium# )l, =H 6 Kw1>H? 6 K+0Kg(co-51

>H? 6 -K-K+0Kg(co-51>H?-# by ma,ing a guess value of >H?/#H? >H?0 )l,510=>H? Kw1>H? = K+Kg(co-1>H? =

7K-K+Kg(co-1>H?-5. The correct concentration of >H? is converted into activity to give

pH# i.e.# )tmospheric !uilibrium pH =$og 0>H?fh5.

Web lin0s to spreadsheets for doing the calculations described in this article:

True 2a234%aturation

0%ee xpandedO%at2alc-.htm5

pHs &egression method

Fissolved 2arbon Fioxide# >23-M? Temperature 2orrection of pH0%ee $angelierO%at2alc+.htm5

)tmospheric !uilibrium pH#$angelier "odel

0%ee )tm!uilOpHO$angelier.htm5

)tmospheric !uilibrium pH# xpanded "odel

0%ee )tm!uilOpHOxpanded.htm5

!eferences"

+. &.F. Kun'# ).D. *en# and T.2. Hess# V2ooling Water 2alculations# 2hemical

ngineering# 0)ugust +B


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4. Yohn W. &y'nar# V) Jew Index for Fetermining )mount of 2alcium 2arbonate

%cale Dormed by a Water# Y )mer Water Wor,s )ssoc 0)pril +B

ph values for cooling water systems

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