1 Introduction

Chlorine disinfection is routinely used by water utilities in P. R. China, the United States, and many other countries^[1-3]. The amount of chlorine applied to a disinfection process must be sufficient to compensate the reaction loss and maintain the required residual over the entire disinfection time. The chlorine residual, on the other hand, cannot be too high to produce excess disinfection byproducts (DBPs) and/or make the water taste and smell like chlorine. An upper limit is typically set at below the maximum residual disinfectant level (MRDL) of 4 mg/L Cl₂^[4]. To prevent the microbial regrowth, potable water is also required to maintain the chlorine residual throughout distribution. The RMDL (required minimum disinfectant level), for example, is 0.2 mg/L Cl₂^[4]. At the treatment plant, the bulk decay or reaction with the substances in bulk water dominates the chlorine consumption. Once in the distribution system, the bulk decay continues while the wall decay also occurs (Note: the wall decay, which is caused by "the pipe walls and the biofilm and corrosion products that adhere to them"^[5], is beyond the scope of this study). The bulk decay can be measured independently while the wall decay cannot or has to be determined through subtracting the bulk decay from the total decay. An accurate chlorine bulk decay model will serve to ensure that the potable water meets the microbial and the chemical safeties at the treatment plant and throughout the distribution.

Among the mathematical models available, the general second-order chlorine bulk decay model (GBDM) is most fundamentally sound^{[3, 6-10]}. The GBDM is a multiple channel reaction model that can handle all the reactive substances in water. From the GBDM, various usable models are developed either as direct simplifications (by retaining just one, two or three reactants) or as semi-theoretical variable rate coefficient (VRC) models. Those usable models are accompanied by strong experimental evidences^{[3, 6-11]}. The dispute over which one is better, however, has not been settled. Most noticeably, some researchers^{[6-8, 10]} recommend the two-reactant (2R) model while others^{[3, 9]} question the adequacy of two reactants and favor the semi-theoretical VRC models. To resolve the difference over the possible scenarios, one will need the GBDM and Monte Carlo simulations.

Blends of different waters are increasingly used in area/countries where water sources are scarce. Fisher et al.^[12] recently formulated "a dynamic model of chlorine decay in a blend of two sources" by "linking the two-reactant (2R) models of individual sources". The resulting blend decay model (BDM) has four (4) notional (fictive) reactants. Fisher et al.^[12] demonstrated that their BDM with input parameters determined for individual sources (before blending) can predict the chlorine decay in blends of surface waters, desalinated water, and ground waters. This newly developed BDM is a special case of the GBDM and can be solved analytically as well.

The general second-order chlorine bulk decay model (GBDM) provides a theoretical basis for various usable models^{[3, 6-10]}. The real application of the GBDM, i.e., treating the complex background matter truly as a mixture of many reactive compounds, however, has been hindered by its numerous fictive parameters and lack of an analytical solution. This theoretical work intends to remove the two obstacles and advance the understanding and application of the GBDM. The comprehensive study is organized into three sections. In section 2, we build the GBDM rationally from the individual reactions, solve the GBDM through transformation and integration, and treat the leading usable models as special cases. In section 3, we characterize the reaction heterogeneity with the probabilistic distribution, apply the theory to various scenarios, and revise the analytical solution accordingly. In section 4, we explore the GBDM in various equivalent forms, unify the interpretation of usable models, and establish the basic parameters for assessing/ranking the reactive background matter.

2 GBDM Development

Chlorine decay in potable water is primarily caused by the background natural organic matter (NOM). With the current state of knowledge, it is not infeasible to determine and quantify the NOM compounds individually. The complex NOM is thus treated as a mixture of N number fictive compounds. Each fictive compound is identified with elementary rate coefficient k_i and initial concentration C_{i, 0}. The reaction with chlorine, which is redox in nature, is assumed to be bimolecular and irreversible. By the law of mass action, the rate of reaction is written:

$ {{r}_{i}}=-{{k}_{i}}{{C}_{\text{Cl}}}{{C}_{i}}\left( i=1\ \text{to}\ N \right) $

(1)

where C_i and C_Cl are the concentrations of compound i and chlorine (Cl₂), respectively. The chlorine equivalency, which is a known 1:1 stoichiometric ratio (just like the electron equivalency), is selected and recommended as the measure of all the reactants. The overall reaction rate is obtained as:

$ {{r}_{\text{b}}}=\sum\limits_{i=1}^{N}{{{r}_{i}}}=-{{C}_{\text{Cl}}}\sum\limits_{i=1}^{N}{{{k}_{i}}{{C}_{i}}} $

(2)

The rate of chlorine decay is equal to the overall reaction rate given in Eq.(2). Eqs.(1) and (2) complete the basic rate equations for the general second-order chlorine bulk decay model (GBDM). Evidently, the chlorine decay, which occurs through multiple reaction channels, is modeled with all parallel bimolecular reactions. Any sequential chlorine reaction is assumed to be insignificant or approximated by the parallel reactions. Mathematical justification for representing serial processes with parallel reactions may be found in Forney and Rothman^[13] and Bolker et al.^[14]. Eqs.(1) and (2) can be extended directly to any mixture of compounds regardless of whether they are real, fictive, known or unknown.

Incorporating Eqs.(1) and (2) into the mass balance over a batch or plug-flow reactor results in a form of the GBDM often seen in the literature^{[3, 6-9]}

$ \frac{\text{d}{{C}_{i}}}{\text{d}t}=-{{k}_{i}}{{C}_{\text{Cl}}}{{C}_{i}}\left( i=1\ \text{to}\ N \right) $

(3)

$ \frac{\text{d}{{C}_{\text{Cl}}}}{\text{d}t}=\sum\limits_{i=1}^{N}{\frac{\text{d}{{C}_{i}}}{\text{d}t}}=-{{C}_{\text{Cl}}}\sum\limits_{i=1}^{N}{{{k}_{i}}{{C}_{i}}} $

(4)

$ {{C}_{\text{Cl}}}\left( t=0 \right)={{C}_{\text{Cl,0}}}\ \text{and}\ {{C}_{i}}\left( t=0 \right)={{C}_{i\text{,0}}} $

(5)

where C_{Cl, 0} and C_{i, 0} are the initial concentrations of chlorine and compound i, respectively. Eq.(4) can be integrated and replaced by:

$ {{C}_{\text{Cl,0}}}-{{C}_{\text{Cl}}}=\sum\limits_{i=1}^{N}{\left( {{C}_{i\text{,0}}}-{{C}_{i}} \right)}={{X}_{0}}-X $

(6)

$ {{X}_{0}}=\sum\limits_{i=1}^{N}{{{C}_{i\text{,0}}}} $

(7)

$ X=\sum\limits_{i=1}^{N}{{{C}_{i}}} $

(8)

where X₀ is the initial chlorine demand and X is the (remaining) chlorine demand at reaction time t. With a sufficiently large dose of chlorine: C_{Cl, 0}> X₀, X(t→∞)= 0 and Eq.(6) reduces to

$ {{X}_{0}}={{C}_{\text{Cl,0}}}-{{C}_{\text{Cl}}}\left( t\to \infty \right) $

(9)

where C_Cl(t→∞)> 0. The measurability of X₀ and X, which are the exact amounts of reactive background matter present initially and at time t, further strengths the use of chlorine equivalency with the GBDM. It is also worthwhile to recognize that other measures/units being used in the literature may cause unnecessary complication and/or theoretical inconsistency. For example, the total organic carbon (TOC) is stoichiometrically irresponsible (and might even remain unchanged) during chlorination while the molar ratio is unknown and likely to vary among the NOM compounds.

Eq.(3) is an autonomous system of first order ordinary differential equations (ODEs). Dividing any two of the ODEs with each other eliminates both variables t and C_Cl(t)

$ \frac{\text{d}{{C}_{i}}}{\text{d}{{C}_{j}}}=\frac{{{k}_{i}}}{{{k}_{j}}}\frac{{{C}_{i}}}{{{C}_{j}}}\left( i=1\ \text{to}\ N\ \text{for}\ \text{any}\ j=1\ \text{to}\ N \right) $

(10)

Eq.(10) is integrated:

$ {{C}_{i}}={{C}_{i\text{,0}}}{{\left[ {{C}_{j}}/{{C}_{j\text{,0}}} \right]}^{\left( {{k}_{i}}/{{k}_{j}} \right)}} $

(11)

Subsequently, substituting Eq.(11) into Eqs.(8) and (6) gives:

$ X=\sum\limits_{i=1}^{N}{{{C}_{i\text{,0}}}{{\left[ {{C}_{j}}/{{C}_{j\text{,0}}} \right]}^{\left( {{k}_{i}}/{{k}_{j}} \right)}}} $

(12)

$ {{C}_{\text{Cl}}}={{C}_{\text{Cl,0}}}-{{X}_{0}}+\sum\limits_{i=1}^{N}{{{C}_{i\text{,0}}}{{\left[ {{C}_{j}}/{{C}_{j\text{,0}}} \right]}^{\left( {{k}_{i}}/{{k}_{j}} \right)}}} $

(13)

To relate C_j/C_{j, 0} to reaction time t, Eq.(13) is substituted back into Eq.(3). The resulting ODE is integrated:

$ \int_{1}^{\left( {{C}_{j}}/{{C}_{j\text{,0}}} \right)}{\frac{1/x}{{{C}_{\text{Cl,0}}}-{{X}_{0}}+\sum\limits_{i=1}^{N}{{{C}_{i\text{,0}}}{{x}^{\left( {{k}_{i}}/{{k}_{j}} \right)}}}}\text{d}x}=-{{k}_{j}}t $

(14)

Although it cannot be expressed in elementary functions, the integral can be evaluated to any desired level of accuracy using any of the quadrature formulas^[15].

Either chlorine residual C_Cl(t) or reaction time t(C_Cl) can be predicted with Eq.(13) or Eq.(14). The two equations, which are linked through intermediate variable C_j/C_{j, 0} only, can be solved in sequence. Let

$ y=\frac{{{C}_{j}}}{{{C}_{j\text{,0}}}} $

(15)

To determine time t(C_Cl) for C_{Cl, 0} to reduce to C_Cl, one may first solve Eq.(13) using the Newton-Raphson method

$ {{y}_{+}}={{y}_{c}}-\frac{{{C}_{\text{Cl,0}}}-{{C}_{\text{Cl}}}-{{X}_{0}}+\sum\limits_{i=1}^{N}{{{C}_{i\text{,0}}}y_{c}^{\left( {{k}_{i}}/{{k}_{j}} \right)}}}{\sum\limits_{i=1}^{N}{{{C}_{i\text{,0}}}\left( {{k}_{i}}/{{k}_{j}} \right)y_{c}^{\left( {{k}_{i}}/{{k}_{j}}-1 \right)}}} $

(16)

where y_c is the current estimate and y₊ is the new estimate for y (0≤y≤1). Once a desirable value for y is obtained, t(C_Cl) is computed directly using Eq.(14). Likewise, to predict residual C_Cl(t) after time t, one may first solve Eq.(14) using the Newton-Raphson method

$ \begin{array}{l} {y_ + } = {y_c} - {y_c}\left[ {{C_{{\rm{Cl,0}}}} - {X_0} + \sum\limits_{i = 1}^N {{C_{i{\rm{,0}}}}y_c^{\left( {{k_i}/{k_j}} \right)}} } \right]\left[ {{k_j}t + } \right.\\ \;\;\;\;\;\;\;\left. {\int_1^{{y_c}} {\frac{{1/x}}{{{C_{{\rm{Cl,0}}}} - {X_0} + \sum\limits_{i = 1}^N {{C_{i{\rm{,0}}}}{x^{\left( {{k_i}/{k_j}} \right)}}} }}{\rm{d}}x} } \right] \end{array} $

(17)

and then compute C_Cl(t) using Eq.(13).

The well-known two-reactant (2R) model is developed by retaining just two terms of Eqs.(3) and (4)^{[6-8, 16]}. The expectation is that one reactant would control the initial fast decay while the other reactant dominates the slow decay that follows. The two reactants are named as the fast (F) and the slow (S) reducing agents. From Eqs.(13) and (14) with N= 2, we obtain an analytical solution for the 2R model:

$ {C_{{\rm{Cl}}}} = {C_{{\rm{Cl,0}}}} - \left( {{F_0} + {S_0}} \right) + F + {S_0}{\left( {F/{F_0}} \right)^{\left( {{k_{\rm{S}}}/{k_{\rm{F}}}} \right)}} $

(18)

$ \int_1^{\left( {F/{F_0}} \right)} {\frac{{1/x}}{{{C_{{\rm{Cl,0}}}} - \left( {{F_0} + {S_0}} \right) + {F_0}x + {S_0}{x^{\left( {{k_{\rm{S}}}/{k_{\rm{F}}}} \right)}}}}{\rm{d}}x} = - {k_{\rm{F}}}t $

(19)

where F₀ and S₀ are the initial concentrations, F and S are the concentrations at time t, and k_F and k_S are the rate coefficients. Through an algebraic manipulation, Eqs.(18) and (19) can be shown to be the same as the solution given by Kohpaei and Sathasivan^[7]. The 2R model^{[6-8, 16]}, however, is currently formulated with two deficiencies: 1) retaining just two terms of Eqs.(3) and (4)^{[6-8, 16]} detaches the model from the GBDM; and 2) the F and the S reducing agents are ambiguously defined. In section 4, we will make the necessary clarification and develop/interpret the 2R model more coherently.

Fisher et al.^[12] recently formulated "a dynamic model of chlorine decay in a blend of two sources" "by linking the two-reactant (2R) models of individual sources through weighting the initial concentrations" by dilution factors. This blend decay model (BDM) is also a special case of the general second-order chlorine bulk decay model (GBDM). Let's consider water Sources 1 and 2 with parameters listed in Table 1. The two waters are blended at volume ratio a/(1-a), where a and (1-a) are the volume fractions of Sources 1 and 2. The blend or composite water has four (4) fictive compounds at initial concentrations aF_{1, 0}, aS_{1, 0}, (1-a)F_{2, 0} and (1-a)S_{2, 0}, respectively. From Eqs.(13) and (14) with N= 4, we obtain an analytical solution for the BDM:

$ \begin{array}{l} {C_{{\rm{Cl}}}} = {C_{{\rm{Cl,0}}}} - {X_0} + a\left[ {{F_1} + {S_{1,0}}{{\left( {{F_1}/{F_{1,0}}} \right)}^{\left( {{k_{{\rm{S,1}}}}/{k_{{\rm{F,1}}}}} \right)}}} \right] + \\ \;\;\;\;\;\;\;\;\left( {1 - a} \right)\left[ {{F_{2,0}}{{\left( {{F_1}/{F_{1,0}}} \right)}^{\left( {{k_{{\rm{F,2}}}}/{k_{{\rm{F,1}}}}} \right)}} + } \right.\\ \;\;\;\;\;\;\;\;\left. {{S_{2,0}}{{\left( {{F_1}/{F_{1,0}}} \right)}^{\left( {{k_{{\rm{S,2}}}}/{k_{{\rm{F,1}}}}} \right)}}} \right] \end{array} $

(20)

$ \int_1^{\left( {{F_1}/{F_{1,0}}} \right)} {\frac{{1/x}}{{{C_{{\rm{Cl,0}}}} - {X_0} + a\left[ {{F_{1,0}}x + {S_{1,0}}{x^{\left( {{k_{{\rm{S,1}}}}/{k_{{\rm{F,1}}}}} \right)}}} \right] + \left( {1 - a} \right)\left( {{F_{2,0}}{x^{\left( {{k_{{\rm{F,2}}}}/{k_{{\rm{F,1}}}}} \right)}}} \right) + {S_{2,0}}{x^{\left( {{k_{{\rm{S,2}}}}/{k_{{\rm{F,1}}}}} \right)}}}}{\rm{d}}x} = - {k_{{\rm{F,1}}}}t $

(21)

where X₀ is the initial chlorine demand of the blend

$ {X_0} = a\left( {{F_{1,0}} + {S_{1,0}}} \right) + \left( {1 - a} \right)\left( {{F_{2,0}} + {S_{2,0}}} \right) $

(22)

The general second-order chlorine bulk decay model (GBDM) rationally treats the complex natural organic matter (NOM) with a large (N) number of fictive compounds. The number of fictive parameters grows rapidly at rate (2N+1) (Note: N is often prescribed but itself is a fictive parameter). For example, the simple 2R model of Eqs.(18) and (19) has four parameters while the BDM of Eqs.(20) and (21) has eight parameters. To effectively manage the fictive parameters, a theoretical treatment below is presented.

3 Distribution of Rate Coefficients

Natural organic matter (NOM), formed from biomass degradation through many random reactions and channels, is physically and chemically heterogeneous^[17]. The chemical heterogeneity, although discrete in reality, may be systematically characterized with a continuous distribution. Noticeable examples are the NOM molecular size distribution^[18-19]; the proton and the metal bindings by NOM^{[17, 20-21]}; and the adsorption of NOM by activated carbon^[22-23]. The lognormal distribution, either by itself or as a building block, describes the NOM heterogeneity well. This theoretical treatment is actually quite general and classical, owing to Pauling et al.^[24], Karush and Sonenberg^[25], and others^[26]. Pauling et al.^[24] first applied the lognormal distribution to the intramolecular heterogeneity (functional group variation within same macromolecules). Karush and Sonenberg^[25] later on showed that the same approach was applicable to the intermolecular heterogeneity (functional group variation associated with different molecules).

Reaction heterogeneity of NOM can be both intramolecular and intermolecular. The general second-order chlorine bulk decay model (GBDM) makes no differentiation between the two while treating the NOM with N number fictive compounds. The fictive compounds are identified with elementary reaction rate coefficients [k₀, k₁, …, k_i, …, k_N] complemented by initial concentrations [C_{0, 0}, C_{1, 0}, …, C_{i, 0}, …, C_{N, 0}], where C_{i, 0} matches k_i only and k_i-1 < k_i. An elementary rate coefficient is associated with the functional groups that have the same reaction activation energy (Note: the same functional groups may not necessarily belong to one particular compound). Semi-infinite range [0, ∞) is prescribed for k_i (i=1 to N). This ensures that all the possible rate coefficients are included. A completely-inert compound has no chlorine demand: C_{0, 0}= 0 for k₀= 0; and an instantaneously-reactive compound is unlikely present: C_{N, 0}= 0 as k_N→∞. Let index i vary randomly between points 1 and N. Then, k_i is realized as a discrete random variable with C_{i, 0} as its occurring frequency. The probability for k_i to assume a value from set [k₀, k₁, …, k_i, …, k_N] is:

$ P\left( {{k_i}} \right) = \frac{{{C_{i{\rm{,0}}}}}}{{\sum\limits_{j = 1}^N {{C_{j{\rm{,0}}}}} }} = \frac{{{C_{i{\rm{,0}}}}}}{{{X_0}}} $

(23)

Varying with discrete random k_i (i=1 to N), P(k_i) is a probability mass function

$ P\left( {{k_i}} \right) = \int_{\left( {{k_i} - \Delta k/2} \right)}^{\left( {{k_i} + \Delta k/2} \right)} {f\left( k \right){\rm{d}}k} $

(24)

where f(k) is the probability density function. Provided that difference Δk= (k_i -k_i-1) is small,

$ \int_{\left( {{k_i} - \Delta k/2} \right)}^{\left( {{k_i} + \Delta k/2} \right)} {f\left( k \right){\rm{d}}k} = \frac{{{\rm{d}}\mathit{\Gamma }}}{{{\rm{d}}k}}\left| {_{k = {k_i}}{\rm{d}}k} \right. = f\left( {k = {k_i}} \right){\rm{d}}k $

(25)

where F(k) is the cumulative distribution function. Finally, combining Eqs.(23), (24) and (25) results in

$ \begin{array}{l} \frac{{{C_{i{\rm{,0}}}}}}{{{X_0}}} = F\left( {k = {k_i} + \Delta k/2} \right) - F\left( {k = {k_i} - \Delta k/2} \right) = \\ \;\;\;\;\;\;\;\;f\left( {k = {k_i}} \right){\rm{d}}k \end{array} $

(26)

By matching the probabilities, the discrete distribution of k_i (i= 1 to N) is coherently transformed into a continuous distribution of k and vice versa. The theoretical treatment opens the door for the GBDM to work with a family of probability distributions and reduces the fictive parameters to a minimum.

Chemical heterogeneity of NOM or a class of compounds may follow the law of lognormal distribution^{[13, 17, 19-20, 22-25, 27]}. Eq.(26) then becomes:

$ {C_{i{\rm{,0}}}} = {X_0}\frac{1}{{\sigma \sqrt {2{\rm{\pi }}} }}\frac{1}{{{k_i}}}{{\rm{e}}^{ - {{\left[ {\left( {\ln {k_i} - \ln {k_{\rm{m}}}\sigma \sqrt 2 } \right)/} \right]}^2}}}{\rm{d}}k $

(27)

where k_m is the median rate coefficient, and σ is the variance of lnk distribution. Following Pauling et al.^[24], σ is referred to as the heterogeneity index. A lognormal distribution of rate coefficient k is skewed with many small k values and fewer large ones. The skewness increases with σ,

$ skew = \left( {{{\rm{e}}^{{\sigma ^2}}} + 2} \right)\sqrt {{{\rm{e}}^{{\sigma ^2}}} - 1} $

(28)

Substituting Eq.(27) into Eq.(11) gives:

$ {C_i} = {X_0}{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)^{\left( {{k_i}/{k_{\rm{m}}}} \right)}}\frac{1}{{\sigma \sqrt {2{\rm{\pi }}} }}\frac{1}{{{k_i}}}{{\rm{e}}^{ - {{\left[ {\ln \left( {{k_i}/{k_{\rm{m}}}} \right)/\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}k $

(29)

Subsequently, substituting Eq.(29) into Eqs.(12), (13) and (14) yields:

$ {X = {X_0}\frac{1}{{\sigma \sqrt {2\pi } }}\int_0^\infty {{{\left( {{C_{\rm{m}}}/{C_{{\rm{m}},{\rm{0}}}}} \right)}^\gamma }\frac{1}{\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } } $

(30)

$ \begin{array}{l} {C_{{\rm{Cl}}}} = {C_{{\rm{Cl}},{\rm{0}}}} - {X_0}\left[ {1 - \frac{1}{{\sigma \sqrt {2\pi } }}\int_0^\infty {{{\left( {{C_{\rm{m}}}/{C_{{\rm{m}},{\rm{0}}}}} \right)}^\gamma } \cdot } } \right.\\ \left. {\;\;\;\;\;\;\;\;\frac{1}{\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } \right] \end{array} $

(31)

$ \begin{array}{l} \int_1^{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)} {\frac{{1/x}}{{{C_{{\rm{Cl,0}}}} - {X_0}\left( {1 - \frac{1}{{\sigma \sqrt {2{\rm{\pi }}} }}\int_0^\infty {{x^\gamma }\frac{1}{\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } } \right)}}{\rm{d}}x} = \\ \;\;\;\;\;\;\;\; - {k_{\rm{m}}}t \end{array} $

(32)

Eqs.(31) and (32) are coupled through intermediate variable C_m/C_{m, 0} only. The two equations can thus be solved sequentially, i.e., for C_m(t)/C_{m, 0} and then for C_Cl(t) (or for C_m(C_Cl)/C_{m, 0} and then for t(C_Cl)), very much in the same way shown earlier for solving Eqs.(13) and (14).

Heterogeneity index σ lies somewhere in the range of 0 ≤ σ < ∞. As σ→ 0, the Gaussian $ \delta \left( \eta \right) = \frac{1}{{\sigma \sqrt {2\pi } }}{{\rm{e}}^{- {{[\eta /(\sigma \sqrt 2 )]}^2}}} $ becomes a Dirac delta function as δ(η)=0 for η≠0 and $ \int_{-\infty }^\infty {\delta \left( \eta \right){\rm{d}}} \eta = 1$. The integral in Eq.(32) approaches to x

$ \begin{array}{l} \mathop {\lim }\limits_{\sigma \to 0} \frac{1}{{\sigma \sqrt {2{\rm{\pi }}} }}\int_0^\infty {{x^\gamma }\frac{1}{\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } = \mathop {\lim }\limits_{\sigma \to 0} \int_{ - \infty }^\infty {{x^{{{\rm{e}}^\eta }}} \cdot } \\ \;\;\;\;\;\frac{1}{{\sigma \sqrt {2{\rm{\pi }}} }}{{\rm{e}}^{ - {{\left[ {\eta /\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\eta = x \end{array} $

(33)

Likewise, the integral in Eq.(31) converges to (C_m/C_{m, 0}) as σ→ 0. Eqs.(32) and (31) reduce to:

$ \int_1^{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)} {\frac{1}{{x\left[ {{C_{{\rm{Cl,0}}}} - {X_0}\left( {1 - x} \right)} \right]}}{\rm{d}}x} = - {k_{\rm{m}}}t $

(34)

$ {C_{{\rm{Cl}}}} = {C_{{\rm{Cl,0}}}} - {X_0}\left[ {1 - \left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)} \right] $

(35)

where C_{m, 0}= X₀ and C_m=X satisfy the conservation of mass. Evaluating the integral yields:

$ {C_{{\rm{Cl}}}} = \frac{{{C_{{\rm{Cl,0}}}} - {X_0}}}{{1 - \left( {{X_0}/{C_{{\rm{Cl,0}}}}} \right){{\rm{e}}^{{k_{\rm{m}}}\left( {{X_0} - {C_{{\rm{Cl,0}}}}} \right)t}}}}\left( {{C_{{\rm{Cl,0}}}} \ne {X_0}} \right) $

(36)

$ {C_{\mathit{Cl}}} = \frac{{{C_{{\rm{Cl,0}}}}}}{{1 + {C_{{\rm{Cl,0}}}}{k_{\rm{m}}}t}}\left( {{C_{{\rm{Cl,0}}}} = {X_0}} \right) $

(37)

Eqs.(36) and (37) conform to the analytical solution of the second-order single reaction bulk decay model^[1]. The use of lognormal distribution is therefore valid even when there is only one reactive compound. As σ→∞, $ \delta \left( k \right) = \frac{1}{{\sigma \sqrt {2\pi } }}\frac{1}{k}{{\rm{e}}^{{{[{\rm{ln}}\left( {k/{k_{\rm{m}}}} \right)/(\sigma \sqrt 2 )]}^2}}} $ is also a Dirac delta function since δ(k)= 0 for k≠ 0 and $ \int_0^\infty {\delta \left( k \right){\rm{d}}k = {\rm{ }}1} $. The corresponding integral approaches to 1

$ \mathop {\lim }\limits_{\sigma \to 0} \frac{1}{{\sigma \sqrt {2{\rm{\pi }}} }}\int_0^\infty {{{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)}^\gamma }\frac{1}{\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } = 1 $

(38)

Eqs.(30) and (31) then reduce to:

$ X\left( t \right) = {X_0}\;{\rm{and}}\;{C_{{\rm{Cl}}}}\left( t \right) = {C_{{\rm{Cl,0}}}} $

(39)

In summary, the water or background matter is homogeneously reactive if σ= 0, heterogeneously reactive when 0 < σ < ∞, and homogeneously inactive as σ→∞.

A more complicated or composite distribution of rate coefficients may occur when the water is a blend of several sources and/or has some known (organic and/orinorganic) reducing species. We may characterize each water source with rate coefficient distribution Eq.(27), treat the known reducing species as outliers, and combine the individual contributions through superposition. Let′s take a blend of two waters as an example. Source C has initial chlorine demand X_{0, 1} and rate coefficient distribution LN₁(k_m, σ₁²) and Source B has initial chlorine demand X_{0, 2} and rate coefficient distribution LN₂(β_m, σ₂²). After blending Sources C and B at volume ratio a/(1-a), the blend or composite water has initial chlorine demand X₀

$ {X_0} = a{X_{0,1}} + \left( {1 - a} \right){X_{0,2}} $

(40)

and a bimodal lognormal (BLN) distribution of rate coefficients (as a linear combination of LN₁(k_m, σ₁²) and LN₂(β_m, σ₂²)). Applying Eqs.(11) and (27) yields:

$ {C_i} = {\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)^{\left( {{k_i}/{k_{\rm{m}}}} \right)}}\frac{{a{X_{0,1}}}}{{{\sigma _1}\sqrt {2{\rm{\pi }}} }}\frac{1}{{{k_i}}}{{\rm{e}}^{ - {{\left[ {\ln \left( {{k_i}/{k_{\rm{m}}}} \right)/\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}k $

(41)

$ {B_i} = {\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)^{\left( {\beta /{k_{\rm{m}}}} \right)}}\frac{{\left( {1 - a} \right){X_{0,2}}}}{{{\sigma _2}\sqrt {2{\rm{\pi }}} }}\frac{1}{{{\beta _i}}}{{\rm{e}}^{ - {{\left[ {\ln \left( {{\beta _i}/{\beta _{\rm{m}}}} \right)/\left( {{\sigma _2}\sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\beta $

(42)

where C_m/C_{m, 0} is the relative concentration that matches rate coefficient k_m. After summation, the total remaining amounts are obtained:

$ {X_1} = \frac{{a{X_{0,1}}}}{{{\sigma _1}\sqrt {2{\rm{\pi }}} }}\int_0^\infty {{{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)}^\gamma }\frac{1}{\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {{\sigma _1}\sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } $

(43)

$ {X_2} = \frac{{\left( {1 - a} \right){X_{0,2}}}}{{{\sigma _2}\sqrt {2{\rm{\pi }}} }}\int_0^\infty {{{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)}^{\left( {{\beta _{\rm{m}}}/{k_{\rm{m}}}} \right)\gamma }}\frac{1}{\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {{\sigma _2}\sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } $

(44)

Finally, Eqs.(43) and (44) are substituted into Eqs.(13) and (14) to give:

$ {C_{{\rm{Cl}}}} = {C_{{\rm{Cl,0}}}} - {X_0} +\\ \int_0^\infty {\left[ {\frac{{a{X_{0,1}}}}{{{\sigma _1}\sqrt {2{\rm{\pi }}} }}{{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)}^\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {{\sigma _1}\sqrt 2 } \right)} \right]}^2}}} + \frac{{\left( {1 - a} \right){X_{0,2}}}}{{{\sigma _2}\sqrt {2{\rm{\pi }}} }}{{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)}^{\left( {{\beta _{\rm{m}}}/{k_{\rm{m}}}} \right)\gamma }}{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {{\sigma _2}\sqrt 2 } \right)} \right]}^2}}}} \right]} \frac{1}{\gamma }{\rm{d}}\gamma $

(45)

$ - {k_{\rm{m}}}t = \int_1^{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)} {\frac{{1/x}}{{{C_{{\rm{Cl,0}}}} - {X_0} + \int_0^\infty {\left[ {\frac{{a{X_{0,1}}}}{{{\sigma _1}\sqrt {2{\rm{\pi }}} }}{x^\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {{\sigma _1}\sqrt 2 } \right)} \right]}^2}}} + \frac{{\left( {1 - a} \right){X_{0,2}}}}{{{\sigma _2}\sqrt {2{\rm{\pi }}} }}{x^{\left( {{\beta _{\rm{m}}}/{k_{\rm{m}}}} \right)\gamma }}{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {{\sigma _2}\sqrt 2 } \right)} \right]}^2}}}} \right]\frac{1}{\gamma }{\rm{d}}\gamma } }}{\rm{d}}x} $

(46)

The two equations, also linked through intermediate variable C_m/C_{m, 0} only, can be solved sequentially for C_m(t)/C_{m, 0} first and then for C_Cl(t) (or for C_m(C_Cl)/C_{m, 0} and then for t(C_Cl)).

The general second-order chlorine bulk decay model (GBDM)originally has two basic equations: rate equations Eqs.(1) and (2). Here we recommend that rate coefficient distribution Eq.(26) or (27) be the third basic equation. Given that potable water is typically dominated by background natural organic matter (NOM), a rational approach to implement the GBDM is to adopt the common lognormal distribution Eq.(27), measure the chlorine decay in batch reactors, and calibrate the model using the analytical solution Eqs.(29) through (31). Once initial chlorine demand X₀, median rate coefficient k_m, and heterogeneity index σ are known, the GBDM in Eqs.(1), (2), and (27) can be applied to any reaction systems, such as disinfection contactors, clear wells/storage tanks, and distribution networks.

4 Mean Reaction Rate Coefficient(s)

The general second-order chlorine bulk decay model (GBDM) given by Eqs.(3) and (4) can be rearranged into a form that explicitly matches the overall reaction ^{[3, 9-10]}

$ \frac{{{\rm{d}}{C_{{\rm{Cl}}}}}}{{{\rm{d}}t}} = \frac{{{\rm{d}}X}}{{{\rm{d}}t}} = - \kappa {C_{{\rm{Cl}}}}X $

(47)

$ \kappa = \frac{1}{X}\sum\limits_{i = 1}^N {{k_i}{C_i}} = \sum\limits_{i = 1}^N {\left( {{C_i}/\sum\limits_{j = 1}^N {{C_j}} } \right){k_i}} $

(48)

Rate coefficient κ maintains several physical significances while varying with reaction time t∈[0, ∞). Firstly, κ is the rate coefficient of the overall reaction and is equal to the rate coefficient of chlorine decay. Secondly or furthermore, κ is the mean reaction rate coefficient associated with the functional group distribution of background matter. By differentiating Eq.(48), Jonkergouw et al.^[3]and Hua et al.^[9] showed:

$ \frac{{{\rm{d}}\kappa }}{{{\rm{d}}t}} < 0\left( {{\rm{if}}\;N \ge 2} \right)\;{\rm{and}}\;\frac{{{\rm{d}}\kappa }}{{{\rm{d}}t}} = 0\left( {{\rm{if}}\;N = 1} \right) $

(49)

i.e., the multiple channel reactions (N≥2) cause the mean reaction rate coefficient to decrease monotonically. The conceptual and mathematical deduction coincides with a long standing observation that the chlorine decay in potable water has a variable or continuously-declining rate coefficient.

After substituting Eqs.(11) and (12) into Eq.(48), an analytical solution for the mean reaction rate coefficient is obtained:

$ \kappa = \frac{{\sum\limits_{i = 1}^N {{k_i}{C_{i,0}}{{\left( {{C_j}/{C_{j,0}}} \right)}^{\left( {{k_i}/{k_j}} \right)}}} }}{{\sum\limits_{i = 1}^N {{C_{i,0}}{{\left( {{C_j}/{C_{j,0}}} \right)}^{\left( {{k_i}/{k_j}} \right)}}} }} $

(50)

where C_j(t)/C_{j, 0} is given by Eq.(14). The initial chlorine decay rate coefficient is equal to:

$ {\kappa _0} = \kappa \left( {t = 0} \right) = \frac{1}{{{X_0}}}\sum\limits_{i = 1}^N {{k_i}{C_{i,0}}} = \sum\limits_{i = 1}^N {\left( {{C_{i,0}}/\sum\limits_{j = 1}^N {{C_{j,0}}} } \right){k_i}} $

(51)

For the lognormal distribution of functional groups, Eqs.(29) and (30) are substituted into Eq.(48) directly:

$ \kappa = \frac{{{k_m}\int_0^\infty {{{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)}^\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } }}{{\int_0^\infty {{{\left( {{C_{\rm{m}}}/{C_{{\rm{m,0}}}}} \right)}^\gamma }\frac{1}{\gamma }{{\rm{e}}^{ - {{\left[ {\ln \gamma /\left( {\sigma \sqrt 2 } \right)} \right]}^2}}}{\rm{d}}\gamma } }} $

(52)

where C_j(t)/C_{j, 0} is given by Eq.(32). Initially (t=0),

$ {\kappa _0} = \kappa \left( {t = 0} \right) = {k_m}{{\rm{e}}^{\frac{{{\sigma ^2}}}{2}}} $

(53)

which shows that initial chlorine decay rate coefficient κ₀ is directly correlated with median rate coefficient k_m and heterogeneous index σ. When σ =0, κ₀= k_m, when 0 < σ < ∞, κ₀> k_m, and as σ→∞, $ \mathop {\lim }\limits_{\sigma \to \infty } \left( {\frac{{{\kappa _0}}}{{{k_{\rm{m}}}}}} \right) \to \infty $ where both κ₀ → 0 and k_m →0.

Significantly, initial chlorine decay rate coefficient κ₀ is equal to the mean reaction rate coefficient associated with the functional group distribution of background water in original water. For any water or functional group distribution, κ₀ can be determined directly from the initial chlorine decay

$ {\kappa _0} = \frac{1}{{{C_{{\rm{Cl,0}}}}{X_0}}}\left( { - \frac{{{\rm{d}}{C_{{\rm{Cl}}}}}}{{{\rm{d}}t}}\left| {_{t \to 0}} \right.} \right) = \frac{1}{{{C_{{\rm{Cl,0}}}}{X_0}}}\left( {\frac{{{C_{{\rm{Cl,0}}}} - {C_{{\rm{Cl}}}}\left( {\Delta t} \right)}}{{\Delta t}}} \right)\left| {_{t \to 0}} \right. $

(54)

where X₀ is measurable according to Eq.(9). The conceptual connection and measurability makes κ₀ a unique kinetic parameter for accessing/ranking the background NOM (which varies from water to water).

The reactive background matter collectively controls the chlorine decay through its mean reaction rate coefficient. This mechanism naturally extends to any fraction of the background matter. If the background matter is split into slow (S) and fast (F) reactive fractions, for example,

$ {k_{{\rm{ave,S}}}} = \frac{1}{m}\sum\limits_{i = 1}^m {{k_i}} \;{\rm{and}}\;{k_{{\rm{ave,F}}}} = \frac{1}{{\left( {N - m} \right)}}\sum\limits_{i = m + 1}^N {{k_i}} $

(55)

$ {S_0} = \sum\limits_{i = 1}^m {{C_{i,0}}} \;{\rm{and}}\;S = \sum\limits_{i = 1}^m {{C_i}} $

(56)

$ {F_0} = \sum\limits_{i = m + 1}^N {{C_{i,0}}} \;{\rm{and}}\;F = \sum\limits_{i = m + 1}^N {{C_i}} $

(57)

the general second-order chlorine bulk decay model (GBDM) takes on the form

$ \frac{{{\rm{d}}C}}{{{\rm{d}}t}} = \frac{{{\rm{d}}X}}{{{\rm{d}}t}} = \frac{{{\rm{d}}F}}{{{\rm{d}}t}} + \frac{{{\rm{d}}S}}{{{\rm{d}}t}} $

(58)

$ \frac{{{\rm{d}}F}}{{{\rm{d}}t}} = - {k_F}{C_{{\rm{Cl}}}}F $

(59)

$ \frac{{{\rm{d}}S}}{{{\rm{d}}t}} = - {k_S}{C_{{\rm{Cl}}}}S $

(60)

$ {k_S} = \frac{1}{S}\sum\limits_{i = 1}^m {{k_i}{C_i}} = \sum\limits_{i = 1}^m {\left( {{C_i}/\sum\limits_{j = 1}^m {{C_j}} } \right){k_i}} $

(61)

$ {k_F} = \frac{1}{F}\sum\limits_{i = m + 1}^N {{k_i}{C_i}} = \sum\limits_{i = m + 1}^N {\left( {{C_i}/\sum\limits_{j = m + 1}^N {{C_j}} } \right){k_i}} $

(62)

where elementary reaction rate coefficients k_i-1 < k_i, k_{ave, F} and k_{ave, S} are the arithmetic averages, k_F(t) and k_S(t) are the concentration (population) weighted means, and integer m may assume any value between 1 and (N-1). Given that k_{i, 1} < k_i, k_{ave, F} > k_{ave, S} is absolutely true. However, k_{ave, F} > k_{ave, S} guarantees neither k_F(t=0)> k_S(t=0) nor that the F fraction controls the initial chlorine decay. At any reaction time t∈[0, ∞), the relative contribution of the two fractions is governed by:

$ \frac{{{\rm{d}}F}}{{{\rm{d}}S}} = \frac{{\left( {\frac{{{\rm{d}}F}}{{{\rm{d}}t}}} \right)}}{{\left( {\frac{{{\rm{d}}S}}{{{\rm{d}}t}}} \right)}} = \frac{{{k_F}F}}{{{k_S}S}} $

(63)

For the F fraction to dominate the initial chlorine decay, condition F₀k_F(t=0)>>S₀k_S(t=0) must be met. If F(t)k_F(t) and S(t)k_S(t) are comparable, both F and S fractions contribute significantly to the chlorine decay. When F(t) k_F(t) < < S(t) k_S(t), on the other hand, the S fraction takes over the control of chlorine decay. In all the circumstances, mean rate coefficients, k_F and k_S continuously decline with reaction time t. Only when k_S and k_F are invariable with t, Eqs.(55) through (62) reduce to the existing two-reactant (2R) model. Conceptually, though, k_S and k_F defined by Eqs.(61) and (62) are the mean rate coefficients of S and F fractions while k_S and k_F in the existing 2R model ^{[6-8, 16]} are the rate coefficients of single reactants. There are (N-1) number ways to split the background matter into two fractions. Also the background mater can be divided into a maximum of N number fractions. Therefore, the GBDM may take on numerous equivalent forms. Those equivalent forms are the most logical basis for simplifying or approximating the GBDM. This significant aspect has been unfortunately overlooked by the usable models^{[1, 6-8, 16]} that retain just one, two or a few terms of Eqs.(1) and (2).

Jonkergouw et al.^{[3, 10]} and Hua et al.^[9] recently proposed the semi-theoretical variable rate coefficient (VRC) models. Their idea is to approximate the overall reaction rate coefficient (κ) with a function of chlorine demand (X/X₀). Jonkergouw et al.^[3] started with:

$ \frac{{{\rm{d}}\kappa }}{{{\rm{d}}t}} = {\alpha _J}{C_{{\rm{Cl}}}}\left( {{k_{\min }}\kappa - {\kappa ^2}} \right) $

(64)

where α_J>0 is an empirical parameter and k_min is the smallest rate coefficient. After dividing Eq.(64) with Eq.(47), the differential equation is solved with initial condition κ(t=0) = κ₀

$ \begin{array}{l} \kappa = \left( {{\kappa _0} - {k_{\min }}} \right){\left( {X/{X_0}} \right)^{{\alpha _J}}} + {k_{\min }} = \left( {{\kappa _0} - {k_{\min }}} \right)X_0^{ - {\alpha _J}} \cdot \\ \;\;\;\;\;\;{\left( {{X_0} - {C_{{\rm{Cl,0}}}} + {C_{{\rm{Cl}}}}} \right)^{{\alpha _J}}} + {k_{\min }} \end{array} $

(65)

Now it is clear that the Jonkergouw et al.'s VRC model^{[3, 10]} approximates κ with a power function. Substituting Eq.(65) into Eq.(47) gives:

$ \frac{{{\rm{d}}{C_{{\rm{Cl}}}}}}{{{\rm{d}}t}} = \frac{{{\rm{d}}X}}{{{\rm{d}}t}} = - \left[ {\left( {{\kappa _0} - {k_{\min }}} \right){{\left( {X/{X_0}} \right)}^{{\alpha _J}}} + {k_{\min }}} \right]{C_{{\rm{Cl}}}}X $

(66)

Eq.(66) is solved analytically

$ \int_{{C_{{\rm{Cl,0}}}}}^{{C_{{\rm{Cl}}}}} {\frac{{1/x}}{{\left\{ {\left( {{X_0} - {C_{{\rm{Cl,0}}}} + x} \right)\left[ {\left( {{\kappa _0} - {k_{\min }}} \right)X_0^{ - {\alpha _J}}{{\left( {{X_0} - {C_{{\rm{Cl,0}}}} + x} \right)}^{{\alpha _J}}} + {k_{\min }}} \right]} \right\}}}{\rm{d}}x} = - t $

(67)

In contrast, the Hua et al.'s VRCmodel^[9] approximates κ directly with an exponential function:

$ \kappa = {\kappa _0}{{\rm{e}}^{ - \left( {{\beta _H}/{X_0}} \right)\left( {{C_{{\rm{Cl,0}}}} - {C_{{\rm{Cl}}}}} \right)}} = {\kappa _0}{{\rm{e}}^{ - {\beta _H}\left( {1 - X/{X_0}} \right)}} $

(68)

where β_H>0 is an empirical parameter. Substituting Eq.(68) into Eq.(47) gives:

$ \frac{{{\rm{d}}{C_{{\rm{Cl}}}}}}{{{\rm{d}}t}} = \frac{{{\rm{d}}X}}{{{\rm{d}}t}} = - {\kappa _0}{{\rm{e}}^{ - {\beta _H}\left( {1 - X/{X_0}} \right)}}{C_{{\rm{Cl}}}}X $

(69)

Eq.(69) is solved analytically as well

$ \int_{{C_{{\rm{Cl,0}}}}}^{{C_{{\rm{Cl}}}}} {\frac{{1/x}}{{\left( {{X_0} - {C_{{\rm{Cl,0}}}} + x} \right){{\rm{e}}^{ - \left( {{\beta _H}/{X_0}} \right)\left( {{C_{{\rm{Cl,0}}}} - x} \right)}}}}{\rm{d}}x} = - {\kappa _0}t $

(70)

5 Conclusions

The general second-order chlorine bulk decay model (GBDM) is clarified based on the concept of fictive compounds, the chlorine equivalency, and the law of mass action. It turns out that the GBDM may take on numerous equivalent forms. The original or basic formulation accentuates the elementary reactions. All the other forms can be derived or interpreted through the mean reaction rate coefficient(s). A suite of analytical solutions are developed for batch and plug flow reactors. Leading usable models in the literature are treated as special cases and solved analytically as well. The analytical solutions provide deep insights into the GBDM and can also facilitate the parameterization and sensitivity analysis.

The background natural organic matter (NOM) is systematically characterized with the probabilistic distribution of functional groups. The theoretical treatment reduces the fictive parameters to a minimum. For the common lognormal distribution, the GBDM needs only three parameters, well defined as initial chlorine demand X₀, median rate coefficient k_m, and heterogeneity index σ. The GBDM is valid over the entire range of σ= 0 to σ→∞ even though the log normal distribution is limited to 0 < σ < ∞. The NOM or background matter is homogeneously reactive when σ= 0; heterogeneously reactive when 0 < σ < ∞; and homogeneously inactive as σ→∞. For more complicated scenarios, composite distributions can be constructed through superposition of individual distributions. A highlighted example is to predict chlorine decay in blends of different waters.

The initial chlorine decay rate coefficient (κ₀) is equal to the mean reaction rate coefficient associated with the functional group distribution of original background mater in water. A simple formula is developed to determine κ₀ directly from the initial chlorine decay. The conceptual connection and measurability makes κ₀ a unique parameter for accessing/ranking the background NOM. With the initial rate coefficient (κ₀) and the initial chlorine demand (X₀) measured, the number of fictive parameters are reduced by two, regardless of the functional group distribution. For the lognormal distribution, κ₀ is correlated directly with median rate coefficient k_m and heterogeneous index σ. The distribution can be described by pairing either k_m and σ or κ₀ and σ. It is natural for the GBDM to use k_m and σ as inputs but it is recommended that the κ₀ be reported explicitly next to the k_m.

Equipped with the concept and formula of functional group distribution, the general second-order chlorine bulk decay model (GBDM) can be applied effectively to any reactive background matter. Future work will focus on characterizing various waters with the GBDM, quantifying the pH and the temperature effects, and integrating the GBDM into the disinfection process and the distribution network models.