Modelling and Simulation in Engineering

Volume 2019, Article ID 5863905, 11 pages

https://doi.org/10.1155/2019/5863905

## A Mathematical Model for Variable Chlorine Decay Rates in Water Distribution Systems

^{1}University of Eswatini, Department of Environmental Health Science, P.O. Box 369, Mbabane, Eswatini^{2}University of Eswatini, Department of Chemistry, Private Bag 4, Kwaluseni, Eswatini

Correspondence should be addressed to Ababu T. Tiruneh; moc.oohay@etubaba

Received 26 February 2019; Accepted 8 May 2019; Published 23 May 2019

Academic Editor: Jing-song Hong

Copyright © 2019 Ababu T. Tiruneh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A model for relating the time-dependent variable rate of reaction to the decay of chlorine residual in water is developed based on the initial chlorine dose, molar concentrations of reactants, and the rate constant itself. The mathematical model, while retaining its second order nature, simplifies the solution as the residual chlorine and aggregate parameters such as molar concentration of reactants can be estimated. The model is based on molar-averaged reaction rates involving arithmetic and harmonic means of reactants that eliminate the individual reaction rates that are difficult to determine. Part of the mathematical assumption used in the derivation of the equations using molar averaging is tested for its validity through theoretical as well as Monte Carlo simulation of the error term over wide ranges of assumed reaction rates and molar concentration of reactants. The second-order variation of the rate of reaction with respect to the initial chlorine concentration has been verified through experimental tests of bulk chlorine decay carried out at different chlorine doses.

#### 1. Introduction

Disinfection of water using chlorine is an essential water treatment process step that renders the water supplied to consumers bacteriologically safe [1]. The amount of chlorine added for disinfection is controlled in such a way that there is an adequate residual chlorine present as the water flows through the distribution system until it reaches the consumers. The presence of a minimum residual chlorine guarantees that recontamination does not occur as well as deterioration in the aesthetic quality of water due to growth of organisms within the distribution system does not occur. Excess chlorination is chlorination of water over and above the minimum dose that is required to ensure potable water quality. Practices of excess chlorination give rise to the formation of disinfection by-products (DBPs) in water, which are compounds known to be associated with health risks related to cancer [2].

The problem related to maintenance of residual chlorine is increased by water supply interruptions that lead to formation of stagnant water, which, during resumption of supply, may be drawn with no chlorine residual left as the water reaches the consumers [3]. Old water supply systems with pipes that have deteriorated linings encourage microbial growth in the distribution pipe, which in turn results in rapid loss of chlorine due to the wall decay reaction [4].

Chlorine is added for disinfection of water in an optimal way depending on the water quality of the treated water prior to disinfection due to the presence of reactants that exert chlorine demand in the bulk water as the water travels through the distribution system. In addition, the dosage of chlorine is also influenced by chlorine consuming reactants present on the walls of the water supply distribution pipes. An optimum dosage is desirable that addresses these consumptions of chlorine in such a way that a low dosage is avoided that does not ensure adequate disinfection of the water supplied while at the same time minimising the instances of excess chlorine that encourage the formation of undesirable disinfection by-products such as trihalomethanes [5].

The reduction of chlorine residual in water supply system take place in the bulk water as well as the walls of pipes and surfaces such as storage tanks present along the distribution system. The bulk decay is a volume-based reduction process whereas the wall decay is a surface area based process. In addition, the nature of reaction and the types of reactants involved in these two processes are different. Water quality models for the reduction of chlorine, therefore, require separate steps for determination of the wall and bulk decay coefficients [5]. The bulk decay is determined by the laboratory batch test on sample of water taken from the water treatment system ready for disinfection. The wall decay rate is often determined by a calibration process as a difference between the chlorine consumption observed in the distribution system and the chlorine consumption due to bulk decay alone determined by laboratory tests [6]. The wall decay depends on pipe conditions including the materials from which pipes are made [7]. In general, laboratory and pilot tests alone cannot adequately represent the chlorine decay process in the actual distribution system. Therefore, the decay model parameters in water quality modeling programs such as the EPANET need to be calibrated against actual observation of chlorine residuals within the distribution systems [8].

Because of the greater domain of space and time through which chlorine residual need to be determined within the distribution system, monitoring of chlorine residuals based on laboratory determination alone is impractical and cost-ineffective [9]. Mathematical models have been developed that trace the decay of chlorine using conservation of mass equation along the travel path of the water in the distribution system. The conservation of mass equation in space and time takes into account the transport of chlorine with the bulk water (advection) and its reduction in the bulk water as well as along the pipe walls as the water travels through the distribution system pipes [10].

Traditional chlorine decay models are simple first-order reactions [2, 3]. The reaction rate constant in the traditional first-order model is an apparent rate constant in which the molar concentration of the reactants exerting chlorine demand is implicitly represented. Because of this, the reaction rate constant changes when the nature of reactants such as the concentration of dissolved organic matter changes [11]. The rate of reaction is also variable based on the concentration of chlorine that is initially applied [12]. The rate of reaction is also influenced by the presence of different reactants with heterogeneous kinetics of reaction. Initially, chlorine is consumed faster by fast reacting species such as organic matter and by species having greater molarity. This is followed by slowly reacting species and species with lower molar concentration present in the chlorinated water.

In an attempt to take into account the effect of different reactants as well as concentration of chlorine applied, a number of different models have been proposed that more or less deviate from the implicit first-order decay rate model. One such model is a parallel two-reaction decay rate involving fast and slow reactants [13]. A number of other variations of such models are also proposed that are second-order models or a hybrid of first- and second-order models [13–17]. However, few researchers played down the importance deviating from the traditional pseudo-first-order reaction in which they argued that the difference in model results in terms of the free residual chlorine present is not of great practical significance [18–20].

The dependence of the rate of reaction for chlorine decay on the initial concentration of chlorine applied has been cited by several researchers [7, 12, 13, 18, 21]. In addition, researchers have addressed several relevant factors such as temperature and oxidisable organic as well as inorganic matter. The model by Hua et al. [13] integrates these several factors in determining the chlorine residual. Hallam et al. [7] found inverse relation between the initial chlorine dose and the rate of reaction. This relationship has also been confirmed by a number of other researchers [13, 18, 22, 23].

Different researchers have approached the effect of the initial chlorine present and of the different reactants exerting chlorine demand differently. Some researchers such as Fisher et al. [24] assumed two-reaction model in which the two reactions involving fast and slow reactants act in parallel. The model is calibrated against ranges of minimum and maximum initial chlorine concentrations expected. Another model by Kastl et al. [25] tried to determine the model parameters for initial chlorine varying between 1 and 4 mg/L. According to these researchers, the model parameters remain constant in this calibrated range.

The model developed by Phillip et al. [5] is a more explicit second-order time-dependent model. The rate of reaction varies second order in time and is also varying directly with the chlorine concentration present. The proposed solution is a trial and error procedure involving solving two differential equations, namely, the equation involving reduction of chlorine and the time-dependent rate of reaction equation. The boundary value problem is solved using Euler’s method. Four parameters need calibration for this model as demanded by the two differential equations. This is done through the bulk decay data in which the optimal values of the parameters are determined by minimising the error between the model result and actual chlorine residual experimentally observed through evolutionary algorithms.

#### 2. Materials and Methods

This research paper provides outline of the mathematical model for variable rate of decay of chlorine that is also accompanied by laboratory experiment for verification of the second order model with respect to the initial chlorine dose used. The mathematical model considers two cases. In the first case, the molar-averaged reaction rate constant variation with time is modeled. In the second case, the overall reaction rate constant in which the molar concentration of reactants are in-built in the rate constant is modeled.

##### 2.1. Experimental Determination of Chlorine Residual

For the experiment involving determination of chlorine residual at different times and under different initial chlorine dosages used, samples of water that underwent conventional treatment up to and including rapid sand filtration and just prior to disinfection were collected form the Matsapha water treatment plant and were subsequently used in the experimental trials. The samples collected were brought to the Chemistry Laboratory of the University of Eswatini. Different dosages of chlorine were added to multiple samples, and the chlorine residuals were determined at predetermined intervals of time. The method used for determination of residual chlorine was the iodometric titration as stated in Standard Methods for the Examination of Water and Wastewater [26]. The method is based on the principle that when potassium iodide is added to a sample of water containing residual chlorine at pH of 8 or less, the residual chlorine liberates iodine from the potassium iodide and the liberated iodine is titrated with sodium thiosulphate (Na_{2}S_{2}O_{3}) titrant.

In order to make the titration stoichiometric, the pH of the sample is reduced to between 3 and 4 by adding acetic acid. The method has expected a detection limit of 0.04 mg/L if 0.001 N sodium thiosulphate titrant is used. All chemicals used had reagent-grade quality, and fresh solutions were prepared every time the experiment was repeated at different time periods.

Depending upon the concentration anticipated, suitable sample volume was taken so that the titrant sodium thiosulphate (Na_{2}S_{2}O_{3}) volume would not exceed 20 mL and the starch indicator volume also is above 0.2 mL. To the sample volume, 5 mL acetic acid was added followed by 1 g of potassium iodide powder measured on weighing balance. After this, the sample was titrated with sodium thiosulphate (Na_{2}S_{2}O_{3}) of appropriate normality. For samples with low anticipated chlorine concentrations, the titrant was prepared to be of low normality. The titration was continued until the yellow colour was almost disappearing. Then, 1 mL of the starch solution was added until the blue colour disappeared. In order to compensate for the method error, blank titration was also performed using distilled-deionised water and that passed through the same procedure as the one used for the actual sample. A minimum of three repetitions were performed for each determination. After titration, mg/L of chlorine residual present in the sample was determined using the following formula:where *A* is the volume of the titrant used for the sample, *B* is the volume of the titrant used for blank, and *N* is the normality of the Na_{2}S_{2}O_{3} titrant.

##### 2.2. Basis of Mathematical Model

The time varying reaction rate constant equation as is suggested by Phillip et al. [5] is used for the mathematical formulation. According to this equation, the concentration-averaged rate of reaction variation with time is given by the following expression:

The original second-order variation of chlorine decay rate with molar concentration of chlorine and reactants and from which equation (2) was derived is given byIn Equations (2) and (3), *k*_{t} is the molar weighed reaction rate, *C*_{t} is the chlorine residual measured at time *t*, X_{it} is the molar concentration of reactant *i* measured at time *t* and reacting with chlorine, *X*_{T} is the total molarity of the reactants exerting chlorine demand, and *N* is the total number of chlorine consuming reactants present in the water.

Meanwhile, the molar-averaged reaction rate *k*_{t} is defined aswhere *k*_{i} is the individual rate of reaction of reactant *i* having molar concentration *x*_{i} and is exerting chlorine demand. Other symbols are as defined for Equation (2). Equation (2) indicates the second-order nature of the variation of the molar-averaged reaction rate constant *k*_{t} with time. It is also clear from Equation (2) that other factors staying constant, the reaction rate constant *k*_{t} varies as a first-order reaction with respect to the chlorine residual present. The aggregate effect of the presence of chlorine residual and this second-order variation of the rate constant are because as the chlorine concentration increases the rate constant tends to decrease. Similar observations were noted by several researchers through empirical experimental determination of the relationship between the reaction rate constant and the initial chlorine used in the experiments [13, 18, 22, 23]. The rate constant appears to decrease inversely with increase in the chlorine dose used in the experiments. Phillip et al. [5] also argue that the rate constant variation with time always stays negative in Equation (2) as the expression in bracket in this equation can be proven either negative or zero only. In other words, it cannot take positive values.

Equation (2) as developed by Phillip et al. [5] is extended in this paper to provide a formula that relates the time variation of rate of reaction *k*_{t} with chlorine residual present and the molar concentration of reactants. The difficulty of dealing with the indeterminate term in the bracket on the right-hand side of Equation (2) and involving the individual reactants can be resolved through mathematical averaging of the product of the individual reaction rates and the molar concentration of reactants. This averaging is later verified by theoretical formulation of the error term and Monte Carlo simulation of the error of averaging under different values of these product terms, i.e., molar concentration of reactants and individual reaction rates.

Starting with the expression in bracket in Equation (2) and reformulating it such that

Rearranging the right-hand side of the above expression, we get

Now defining the average of *k*_{i}*X*_{it} such that

Since it was already defined in Equation (4) that

Equations (4) and (7) are combined to give

Now returning to Equation (5) once again, we getIn the above expression, (*k*_{i}*X*_{it}/*X*_{it}) is used as a weighing factor. This is also analogous to the weighing of a continuous function, i.e.,

In addition, by using similar averaging method, we get

In this case, 1/*X*_{it} are used as weighing factors.

Overall,

The use of weighted average in Equation (13) above for discrete values can be verified theoretically as well as practically using Monte Carlo simulation using random variation of individual reactants’ rate constants and reactants’ molar concentrations as described below. First, the theoretical basis is explained in the following:

Working a little with the right-hand side expression, we get

Now defining the difference term in molar expression of reactants Δ*x*_{ij}, we get

Substituting the above expression for the *x*_{j} term, we get

The right-hand side of the above expression changes to

Because of symmetry, it is easy to show that

Using this symmetry, the expression is reduced further to

Defining the difference terms Δ*kx* such thatand using this term in the above equation, we get

Therefore,where the error term *E* is defined as

The above error term is summation over second-order difference (i.e., Δ*kx* times Δ*x*) and tends to be small compared with other terms in the equation containing *E*:

The second practical verification is using Monte Carlo simulation of the error term and finding the average as well as bracketing the range of variation of the error term *E*. For this purpose, it is assumed that there were five chlorine demanding reactants with individual reaction rates *k*_{i} varying by a minimum factor of 10 and maximum factor of 1000 and with molar concentrations of reactants *x*_{i} also varying by a minimum factor of 10 and maximum factor of 1000. The Monte Carlo simulation is done for each range of variation 10000 times, with the following results shown in Table 1.