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Development of Mathematical Model for Determining the Quantity of Chlorine
Required for Water T reatment
Article    in  Journal of Applied Scienc es R esearch · August 2010
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Abdullahi Mohammed Evuti
Univ ersiti T eknologi Malaysia
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Univ ersity of Abuja
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Journal of Applied Sciences Research, 6(8): 1002-1007, 2010
© 2010, INSInet Publication
Development of Mathematical Model for Determining the Quantity of Chlorine
Required for Water Treatment.
Abdullahi, Mohammed Evuti a nd Abdulkarim, Bala Isah.
Department of Chemical Engineering, University of Abuja
Abstract: In this paper, mathematical model for determin ing the quantities of chlorine required for water
treatment was developed by considering the inter-re lationship between water quality parameter such as
temperature (T), pH and coliform count s (B) and the quantities of chlorine (Q2) required for treatment.
The developed model equations were found to be. Q2 = 0.001836 0.003836 0.000218 10 13.6782pHBT  
(Kg). Comparison of the results from the simulation of these models and experimental data show a good
prediction with correlation coefficients of 0.9949.
Key words: water, chlorine, modeling, simulation, pH
INTRODUCTION
The importance of water to the survival of man
cannot be over emphasized. Abundant supply of wateror lack of it attracts or dissuades settlements the worldover
[1]. To ensure that this water becomes fit for
human consumption, it requires treatment which isdone in a water treatment plant
[2]. The water works
collect water from its natural source, purify it anddeliver it to the consumers. The sequence of processesinvolved in the treatment of water includes aeration,rapid mixing and flocculation, sedimentation, filtration,liming and disinfection. Disinfection of water is aprocess of making the water free of a disease causingmicro-organisms. Accordi ng to Nikoladze, et al
[3] the
methods of water disinfection can be divided into fourmain groups as follows:a) Thermal methods e.g. boiling.b) Method with the use of strong oxidants e.g.
chlorine, ozone, etc.
c) Physical methods e.g. use of ultra sound,
radioactive irradiation, ultra-violet light etc.
d) Oligodynamic methods: based on the use of noble
metal ions.Among these methods mentioned, the use of
oxidants is the most popular. The oxidants employedfor water disinfection include chlorine, chlorine oxide,ozone, iodine, potassium permanganate, hydrogenperoxide, sodium hypochlorite, and calciumhypochlorite. However, in the practice of waterdisinfection, chlorine, ozone and sodium hypochloriteare preferred. According to
[4], the earliest recorded
use of chlorine directly for water disinfection was onan experimental basis, in connection with filtration at
Louisville, Ky in 1896. It was then employed for ashort period in 1879 in England again on experimentalbasis, to sterilize water distribution mains following atyphoid epidemic. Its first continuous use was inBelgium, beginning in 1902 for the dual objective ofaiding coagulation and making water biologically safe.
A model is simply the mathematical abstraction of
real process
[5]. In other words, it is an ordered set of
mathematical equations, inequalities, charts, graphs andso on and numerical solution to these equations.Process modeling has proven to be extremelysuccessful engineering tool for the design andoptimization of physical, chemical and Biologicalprocesses. It provides an avenue of understandingqualitative and quantitative aspects of the phenomenonof interest. Mathematical modeling is versatile and iswidely used in practice, and is a recognized andvaluable adjunct, and usually a precursor of computersimulation.
The high cost of chemical analysis of water has
necessitated the need for various researches into findingalternative method of determining quantities ofchemical required for water treatment such as the useof model equations which allows quantities of chemicalto be predicted from the existing water qualityparameter data. The result from the study can be usedto predict how changes in qualities of raw water e.g.pH, temperature, alkalinity, turbidity, coliform countsetc. will affect the demand for the various treatmentchemicals (Alum, chlorine, & lime). This can serve asa basis for planning in a water treatment plant.
Corresponding Author: Abdulkarim, Bala Isah, Depart ment of Chemical Engineering, University of Abuja
Email: balisa76@yahoo.com
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J. Appl. Sci. Res., 6(8): 1002-1007, 2010
Modeling process: According to[6], chlorine demand is
affected by the following factors, namely; time ofcontact, pH , temperature, iron and Manganese ions,and Microbial content (Bacteriological counts). Analysishave shown that iron and manganese content of watersupply to Kaduna water treatment plant is negligibleand will have no effect on the chlorine demand. Alsothe residence time of water in the clarifier is longenough for complete action of the chlorine.
Therefore quantity of chlorine can now be
expressed in terms of the other 3 parametersmathematically as
Q
2 = f (pH, Temperature, Bacterial count) (1)
If temperature is represented by T, Bacterial count = B
Equation (1) becomes
Q2 = f (pH, T, B) (2)
Effect of Temperature: According to [2], the
disinfection of chlorine increases with temperature.Therefore, the quantity of chlorine required willdecrease as temperature rises. This can be representedas
(3)
21QT
Introducing constant
(4)  251 QKT
Effect of pH: When chlorine dissociate in water, it
produces HCl and HOCl.HOCl is a strongerdisinfecting species
[1]. Higher pH (alkaline medium)
favours the formation of H+ from HOCl and in acidic
medium the reverse reaction is favoured. Thereforelower pH favour disinfection. This can be representedas follows, (5)
2Qp H
pH is the negative log of hydrogen ion
concentration and introducing constant
( 6)  26 10PHQK
Effect of Bacterial Count: The commonest group is
Escherichia coli, a rod – shaped bacterium and it is themost easily tested for and therefore used as anindicator organism referred to as coliform.
The higher the coliform count, the higher the
quantity of chlorine required for disinfection.
(7)
2QB
Introducing constant:
( 8)27QK B
To correct variations resulting from other factors
not accounted for, a constant (K8) is introduced.
Summing equations (4), (6), (8)
(9)  28 5 6 7110PHQKK K K BT   
The constants are determined using the least square
method [7].
Let Q2i be an observed value of quantity of
chlorine and the predicted value from the model is,
( 1 0 )  85 6 71 10pHi
i
iK KK K BT  
Then the error of prediction E will be given by
(11) 28 5 6 71 10ipH
ii
iEQ i KK K K BT    
And the square of the error is,
(12) 2
2
28 5 6 71 10ipH
ii
iE Qi K K K K BT    
And for all sets of data,
(13) 862
2
25 71 10ipH
i i
iE Qi K K K K BT         
And expanding gives
 2 222 2 2 2 2 2
28 5 8 7 8 21 10 2pHi
i iiEQ i K K K K BK Q iT    
   52 62 72 8 5 8 611 22 1 0 2 2 2 1 0ipH pHi
ii iK Qi K Qi K Qi B KK KKTT       
(14)    87 56 57 6711 2 2 10 2 2 10pH pHi
iiiKK B KK KK B i KK B iTT   
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J. Appl. Sci. Res., 6(8): 1002-1007, 2010
To minimize the error i.e. to obtain those values of c onstants that give best prediction by finding the derivative
of with respect to the constants K8,K5,K6&K7 and equating to zero.2
iE
( 1 5 )  2
82 5 6 8 7
81 22 2 21 0 2pH i
iEK Qi K K i K K B iT K      
( 1 6 )   22 2
52 0 2 3
511 1 1 1 22 2 2 1 0 2pH i
ii i i iEK QK K K B iTT T T T K      
( 1 7 )    22
62 5 7
621 2 10 2 10 2 10 2 10ii i ipH pH pH pH i
iEK Qi K K B iT K       
( 1 8 )  2
2
72 8 8 5 6
71 22 2 2 2 2 1 0pH i
iiEK B i Q i B iK B i K B iK B iK B iT K       
Using MATLAB, the values of the constants K8, K5,
K6 & K7 are obtained and then substituted into equation
(9).
Experimental Methods: The samples are described by
the extent of treatment they have undergone e.g. Rawwater (RW), Aerated water (AW), clarified water(CW), filtered water (FW) a nd treated water. Samples
are taken from different points and the aggregate isfound. They are then analyzed in the laboratory.
Determination of Temperature: The temperatures of
the water samples were determined with the aid ofthermometer, beakers and electrodes. The watersamples were collected in beakers and labeledappropriately to avoid mix-up. The thermometer wasswitched on and the electrode was dipped into thebeaker in turn. After a steady value has been attained,the readings were taken.
Determination of pH: The materials required are
colorimetric comparator, sample cells. Two cells wererinsed with sample water and shaken, one cell wasfilled to mark “B” and 10 drops of indicator wereadded. The second cell was filled to mark “A” andplaced in the comparator on the side marked “blank”slide the colorimeter through the slot in the comparatoruntil the colour in both cells appears identical. Thenumber in the matching colour slot is the pH.Estimation of Coliform Density: The materials required
are culture media, fermentation tubes with inverted gastubes, water samples, and in cubator. Starting with 10ml
of the water sample, 1ml was removed and remaining9ml was diluted with distilled water. The dilutedsample was distributed evenly between 5 tubes whichhave been inoculated with lauryl lactose broth.
The tubes were incubated at 35
oC for 24 hours
after which a change in colour from gold to cloudycolour with gas collected in the tube means bacteriaare present.
To confirm the presence of coliform organisms, a
small amount of the tube content was transferred ontoMC Conkey agar, Eosin ethylene blue agar or brilliantgreen agar in a Petri dish and was covered. Thepresence of coliform is confirmed if microbial culturesare observed after 5days. Concentration of totalcoliform in “most probable number per 100ml”(MPN/100ml) is determined using the poisondistribution as quoted by
[8].
11 2 211 2 2 1 11Pq P qnn n nye ee ea            
y = Probable occurrence of a given result
a = constant for a given set of conditions.n
i = Number size for each dilution
pi = Number of positive in dilution
qi = Number of negative tube in dilution.
λ = Coliform density (Number/ml)
The various values of y are then obtained using computer.
RESULTS AND DISCUSSION
The values of the constants obtained using MATLAB are shown below
K5 = 0.038359Kg 0CK6 = 0.000218Kg
K7 = 13.6782Kg (100ml/MPN), K8 = 0.001836Kg
Substitution of the values of the constants, K5, K6, K7 and K8 into equation (9) give the model for quantity
of chlorine as
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J. Appl. Sci. Res., 6(8): 1002-1007, 2010
20.001836 0.03836 0.000128(10 ) 13.6782 ,( )pHQB K gT   
The model was simulated using another set of data obt ained for the period of 52 weeks, and the results from
the simulation of this model are shown in Appendix B. Comparison between experimental quantities of chlorine
used for treatment and the simulation of the model is shown on figure 1.
Fig. 1: Comparison between model and experimental quantites of chlorine used for trestment
Appendix A: Experimental Data for Qu antity of Chlorine Model.
Weekly average values of pH, Temperature, Coliform count and the Quantity of chlorine.
Weeks Temperature (T) pH Coliform count (B) Quantity of Chlorine
(0C) (MPN/100ml) Q2 (Kg)
1 24.0 7.00 25.00 338.3
2 25.0 7.20 25.00 338.33 25.0 7.20 25.00 338.34 24.0 7.20 25.00 338.35 25.0 7.20 22.00 338.36 24.0 6.80 25.00 338.37 24.0 7.20 25.00 338.38 25.0 7.00 25.00 338.39 25.0 6.80 25.00 338.310 25.0 6.80 25.00 338.311 25.0 7.00 20.00 270.612 25.0 7.20 18.00 245.013 25.0 7.00 16.00 217.014 24.0 7.20 16.00 217.015 25.0 7.00 15.10 217.016 25.0 7.10 16.00 217.017 25.0 7.20 16.00 217.018 25.0 7.10 16.00 217.019 25.0 7.20 13.20 180.020 25.0 7.30 12.30 167.021 25.0 7.20 11.60 157.022 25.0 7.40 12.00 157.023 25.0 7.50 10.00 136.024 25.0 7.20 10.70 145.0
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J. Appl. Sci. Res., 6(8): 1002-1007, 2010
Appendix A: Continue
25 25.0 7.60 9.80 136.0
26 25.0 7.40 7.10 100.027 25.0 7.40 6.20 84.028 25.0 7.40 5.10 71.029 23.0 7.40 5.20 71.030 21.0 7.40 4.50 61.031 25.0 7.30 4.00 54.032 24.0 7.40 4.40 71.033 23.0 7.50 3.60 50.034 21.0 7.40 4.00 54.035 20.0 7.30 4.50 61.036 20.0 7.60 4.50 61.037 20.0 7.60 3.60 50.038 20.0 7.60 3.50 50.039 18.0 7.60 3.50 50.040 18.0 7.40 3.40 50.041 18.0 7.50 3.40 50.042 18.0 7.60 3.50 50.043 18.0 7.60 3.50 50.044 18.0 7.40 3.45 50.045 19.0 7.50 3.40 50.046 18.0 7.60 3.40 50.047 17.0 7.40 3.50 50.048 18.0 7.40 3.40 50.049 18.0 7.60 3.50 50.050 18.0 7.60 3.50 50.051 18.0 7.60 3.45 50.052 18.0 7.50 3.40 50.0
Appendix B: Experimental Data for the Simula tion of Quantity of Chlorine Model
Week Temp
0C pH Coliform count .(MPN/100ml) Q2 Experimantal (kg) Q2 Model (kg)
1 24.0 7.20 16.00 220.00 218.85
2 25.0 7.00 16.00 220.00 218.853 25.0 7.10 16.00 220.00 218.854 24.0 7.20 16.00 220.00 218.855 25.0 7.00 15.00 220.00 205.186 25.0 7.10 16.00 220.00 218.857 24.0 7.00 25.00 335.00 341.968 25.0 7.20 25.00 335.00 341.969 24.0 7.20 25.00 335.00 341.9610 24.0 7.20 25.00 335.00 341.9611 25.0 7.00 25.00 335.00 341.9612 24.0 7.20 25.00 335.00 341.9613 25.0 6.80 25.00 335.00 341.9614 24.0 7.00 25.00 335.00 341.9615 25.0 6.80 25.00 335.00 341.9616 25.0 6.80 25.00 335.00 341.9617 25.0 7.20 22.00 335.00 300.9218 24.0 7.00 20.00 335.00 273.5719 24.0 7.20 18.00 220.00 246.2120 25.0 7.00 16.00 220.00 218.8521 25.0 7.00 16.00 220.00 218.8522 25.0 7.10 15.00 220.00 205.1823 25.0 7.20 13.00 180.00 177.8224 25.0 7.30 12.00 180.00 164.1425 24.0 7.30 10.50 150.00 143.6226 23.0 7.50 7.20 100.00 98.4927 21.0 7.40 6.20 100.00 84.8128 20.0 7.40 5.20 70.00 71.1329 20.0 7.40 5.00 70.00 68.3930 20.0 7.60 4.50 60.00 61.5631 20.0 7.60 4.50 60.00 61.5632 20.0 7.60 4.50 60.00 61.5633 20.0 7.60 4.50 60.00 61.5634 18.0 7.60 3.60 50.00 49.2535 18.0 7.40 3.50 50.00 47.8836 18.0 7.50 3.50 50.00 47.88
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J. Appl. Sci. Res., 6(8): 1002-1007, 2010
Appendix B: Continue
37 18.0 7.60 3.40 50.00 46.51
38 18.0 7.60 3.40 50.00 46.5139 18.0 7.40 3.50 50.00 47.8840 18.0 7.50 3.50 50.00 47.8841 18.0 7.60 3.40 50.00 46.5142 18.0 7.60 3.40 50.00 46.5143 18.0 7.60 3.40 50.00 46.5144 18.0 7.40 3.50 50.00 47.8845 19.0 7.40 3.40 50.00 46.5146 19.0 7.60 3.50 50.00 47.8847 19.0 7.60 3.50 50.00 47.8848 19.0 7.60 3.40 50.00 46.5149 19.0 7.50 3.40 50.00 46.5150 23.0 7.40 5.00 70.00 68.3951 23.0 7.40 5.20 70.00 71.1352 23.0 7.40 5.20 70.00 71.13
Discussion of result: According to Ogunaike [9], the
adequacy of a model depends on how close thepredicted values are to the real situation. Simulationresults of this model show that the model gives a goodprediction of chlorine quantity with a correlationcoefficient of 0.9949. However, some deviations couldbe observed in week 5, the experimental value quantityof chlorine was 200Kg while the value from the modelis 205.18Kg. Also, in weeks 17 and 18 theexperimental value of quantity of chlorine was 355Kgwhile the values from the model are 300.9Kg and273.57Kg respectively. These variations may be due tothe influence of other factors such as organicimpurities, total dissolved oxygen and dissolved metalions which affect the efficiency of chlorine but werenot considered in the model
[2].
Conclusion: From the results it can be concluded that
the quantities of chlorine used are dependent on therelative effect of the individual parameters on the watertreatment process. The model equations for theprediction of quantity of chlorine required for watertreatment was found to be
 2 0.03836 0.03836 / 0.000218 10 13.6782 .pHQT k g    
Comparison of the result from the simulation of themodels and experimental data show a good predictionwith a correlation coefficient of 0.9949REFERENCES
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