Modelling of Escherichia coli removal by a low-cost combined drinking water treatment system

This work presents mathematical modelling of Escherichia coli ( E. coli ) removal by a multi-barrier point-of-use drinking water system. The modelled system is a combination of three treatment stages: ﬁ ltration by geotextile fabric followed by ﬁ ltration and disinfection by silver-coated ceramic granular media (SCCGM) then granular activated carbon (GAC) ﬁ ltration. The presented models accounted for removal mechanisms by each treatment stage. E. coli was modelled as a microbial particle. E. coli inactivation by SCCGM was modelled using the Chick ’ s, Chick-Watson, Collins-Selleck and complete mix system bacterial inactivation kinetic models, which were considered adequately representative for describing the removal. Geotextile removal was modelled using colloidal ﬁ ltration theory (CFT) for hydrosol deposition in ﬁ brous media. The ﬁ ltration removal contributions by the SCCGM and GAC were modelled using CFT for removal of colloidal particles by granular media. The model results showed that inactivation by silver in the SCCGM was the main bacterial removal mechanism. Geotextile and GAC also depicted appreciable removals. The theoretical modelling approach used is important for design and optimization of the multi-barrier system and can support future research in terms of material combinations, system costs, etc. Collector diameter, particle size, ﬁ ltration velocity and contact time were identi ﬁ ed as critical parameters for E. coli removal ef ﬁ ciency.

It is sometimes the only cost-effective option in many rural and suburban areas of developing countries. Although efforts to develop simple yet effective low-cost PoU technologies for rural and suburban areas have intensified globally (Treacy ), challenges still exist (Treacy ). Therefore, there is still need for development and/or optimization of more PoU techniques appropriate to poor communities.
Mathematical modelling may assist in the design and optimization (costs, material combination, etc.) of various PoU systems and can support further research in terms of configuration, flow rate, media combination, and so on while also serving as a decision support tool.
A thorough review of literature showed that modelling of PoU and similar systems for contaminant removal or system optimization has mainly been done on uncombined systems, for example: (i) intermittently operated slow sand filters (Fulazzaky et al. ; Jenkins et al. ), (ii) disinfection using chlorine (Lee & Nam ), (iii) disinfection by natural herbs (Somani & Ingole ), (iv) disinfection by silver or silver-coated materials (Chong et  This paper presents modelling of Escherichia coli removal by a combined drinking water system developed by the authors (Siwila & Brink ) as a contribution to research and development on low-cost PoU drinking water treatment. The combined system consists of three treatment stages: pre-filtration by geotextile fabric followed by filtration and disinfection by SCCGM then GAC filtration (Siwila & Brink ). Each of these steps were modelled as a series of compartments by using specialized theoretical removal mechanisms for each barrier. E. coli was modelled as a microbial colloid or particle as proposed in literature (Tufenkji et

MATERIALS AND METHODS
The general experimental methodology aspects (study setting, design aspects, set up, sampling, testing methods, etc.) are presented in Siwila & Brink (). For instance, the measured E. coli log removal values (Table 4 and Siwila & Brink (). The methodology for the present work primarily presents the mathematical modelling approach for prediction of E. coli removal by the modelled system. The schematic diagram of the combined system that is modelled is given in Figure 1. The system consisted of geotextile fabric for pre-filtration, SCCGM for filtration and disinfection, GAC filtration and a safe storage compartment for treated water. The key system parameters, particularly, those applied directly to the modelling in this paper are included in Table 1.
Sampling was done after at least 7.5 liters of water was passed through the system and at varied flow rates for the first 9 runs and at 2 L/h for the last 3 runs (Siwila & Brink ). The first four filtration runs were done at the maximum obtainable flow rate of 10 L/h (Table 4), while subsequent flow rates were varied from 8 L/h to 2 L/h (Table 4). Varying the flow rate was done to arrive at an optimal flow rate and produce varied contact time, and provide data for the modelling done here.
Thus, flow rates were staggered from the highest obtainable to an optimal 2 L/h where 0 CFU/100 ml for E. coli and fecal coliform in the effluent (>99.99% removal) were consistently achieved.
E. coli removal performance modelling procedure E. coli removal by the combined system was modelled as a series of three compartments. The models were coupled as depicted in Figure 2, whereby the effluent from the geotextile was modelled as the influent to the SCCGM and effluent from the SCCGM was modelled as influent to the GAC. Thus, the effluent of one compartment was modelled as influent to the next (Masters & Ela ). This modelling approach was derived from the works of (i) Metcalf  E. coli removal was calculated using numerical models appropriate to each compartment. Input parameters (Table 1) used in the mathematical calculations were obtained using experimental data and from literature. The modelled removals were then calculated using Equation (1)  The total removal efficiency for each experimental run was calculated using Equation (3) (Tien ) and was then applied to the influent E. coli counts for each run. Computation and    Removal predicted ¼ Predicted removal fraction where: LRV predicted ¼ Predicted log removal value Remembering that the length of the filter bed and the residence time play key roles in determining bacterial removals, contact time between E. coli and silver was estimated using Equation (4), adapted from Metcalf & Eddy (), for each run and flow rate A total of eight combined mathematical models (see Table 3) were tested using combinations of disinfection and filtration modelling approaches as given below. The respective disinfection and filtration modelling approaches alongside the various E. coli removal mechanisms and parameter equations are explained below. Thereafter, the eight combined models as were used in the numerical calculations of this study are summarized in Table 3 and the   associated text just above Table 3.
Modelling E. coli removal by the SCCGM E. coli removal by SCCGM was first modelled using disinfection kinetics in the first four combined models (see Table 3).
The removal was thereby modelled as being only due to bacterial inactivation by the silver coating of the media.
The inactivation by silver was modelled using Chick's,

Collins-Selleck, complete-mix system (CMS) model and the
Chick-Watson bacterial inactivation models (de Moel et al.

;
Metcalf & Eddy ) explained below. Thereafter, E. coli removal contribution by SCCGM filtration (Table 3) was included in the last four combined models using the colloidal filtration theory (CFT) numerical modelling procedure explained under E. coli removal by GAC filtration, but using appropriate SCCGM characteristics.

Chick's and Plug flow model
Assuming that for any length, dx, and throughout the corre- Simplifying Equation (5) gives Equation (6): Applying the following boundary conditions to the SCCGM filter bed: (i) at x ¼ 0; N ¼ N o , and (ii) at x ¼ L; N ¼ N e , and remembering that where: Since the form of Equation (7) (7)) and Chick-Watson model (Equation (11) Solving for N e and remembering that t ¼ V Q , then rearranging gives Equation (9) ) The Collins-Selleck model The Collins-Selleck model Equation (10) where: C ¼ concentration of disinfectant, mg/L; t ¼ contact time (in this study, T ≈ EBCT (de Moel et al. )).

The Chick-Watson model
The Chick-Watson model Equation (11) is a refined version of the Chick's model and emphasizes that time required to achieve a certain inactivation level is related to the disinfec- MWH ) by relating the E. coli removal performance of the GAC column of depth L to the SCE of GAC (Equation (12)). Doing a mass balance on a small differential element and integrating over the entire depth Equation (12) gives Equation (13) (14)). The fraction of particles that actually get captured by a single collector is a product of the SCE (η) and the attachment efficiency (α) (Equations (12) and (13)).
Adding λ 1 and λ bm , we obtain a geotextile filter coefficient (λ) that was used in Equation (23): Assuming further that the geotextile filter has the same porosity and uniform collector size distribution throughout its depth, the filter coefficient (λ) is defined in Equation where: Àδ N is the reduction of the concentration of E. coli as microbial particles passing through a layer of thickness δL. Rearranging the above equation yields Equation (22) (Wakeman & Tarleton ; MWH ).
Representing the influent concentration of the microbial particles by N 0 and integrating Equation (22) with L ¼ 0 (as initial conditions) at filter inlet, we obtained Equation (23) which was then used to estimate the fraction of microbial particle concentration remaining in the effluent. Definitions of the combined system mathematical models Overall, eight combined mathematical models as defined below and summarized in Table 3 (3)) by each stage (Figure 2 and Table 3) were applied successively to the influent counts to get effluent E. coli counts and respective LRVs (Table 4).

Model performance assessment
The E. coli removal performance of each model was assessed using the following statistical techniques ( (ii) and (iii) were tested using models 3 and 8 to test the sensitivity of simulated E. coli removal to each condition or parameter ( Figures 5 and 6). The results of the sensitivity analysis are given and explained below under results and discussion.

RESULTS AND DISCUSSION
Comparison of measured and predicted effluent E. coli removals Predicted E. coli removals were calculated using the coupled models presented above, which are based on the removal mechanisms elaborated on earlier. Figure 4 gives  (Figure 4 and Tables 4 and 5) show that the coupled modelsexcept for models 2 and 6reasonably described the combined E. coli removals by the multi-barrier system. Although models 1, 4, 5, and 8 gave slight underestimations for runs with lower contact time, their predictions were considered satisfactory as also shown by the model performance criteria in Table 5.
Models 3 and 7 gave the closest predictions of E. coli removal values with respect to the measured values, but generally overestimated the LRVs. The appreciable performance by models 1, 3, 4, 5, 7 and 8 signifies they simulated the combined physical and chemical E. coli     (Table 5). Similarly, model 8 was chosen over models 1, 2, 4, 5 and 6 since it also depicted better statistics (Table 5). It can be seen from Figure 5 that, although E. coli removal prediction by SCCGM alone seemed to be a good representation, modelling additional removal by other treatment steps (i.e. geotextile and GAC removals) was still important for the models to be fully representative of the multi-barrier system. Thus, from the results shown in Figure 5, it can be seen that disinfection removal alone could not fully describe the E. coli removals, giving predicted LRVs below measured values for both models.
Effect of contact time, filtration rate, collector diameter and microbial particle (E. coli) size The effect of contact time, filtration rate, collector diameter and microbial particle size on E. coli removal was assessed using models 3 and 8 ( Figure 6). Both models indicated that larger contact time (Figure 6(i)) resulted in higher E. coli removal. Since contact time is dependent on filter media depth and filtration rate (see Equation (4) For instance, as filtration rate increases (Figure 6(ii)), E. coli removal by both models decreases. Therefore, careful optimization of these parameters is expected to enhance E. coli removal performance. It is worth noting that filtration rate is affected by various factors, of which particle size distribution is the key factor (Siwila & Brink ).
Therefore, to optimise contact time it is necessary to not only look at filtration rate but also at factors affecting it, The sensitivity of models 3 and 8 to collector and fiber diameter is shown in Figure 6(iii) and 6(v). The smaller the collector or fiber diameter, the higher the removals ( Figure 6(iii) and 6(v)). The sensitivity of microbial particle (E. coli) size was also assessed (Figure 6(iv) and 6(vi)). The effect of the collector/fiber diameter and microbial particle diameter were assessed by applying the varied particle sizes on the geotextile and GAC CFT models, which are directly affected by particle size. This analysis was done using the input parameters listed in Table 1 but keeping the optimal flow rate (2 L/h) constant. It can be seen from Figure 6(iv) and 6(vi) that the microbial particle diameter having the least removal efficiency by GAC and geotextile in both models is somewhere between 1 and 2 μm. Removal of microbial particles below this range increases with decreasing particle diameter because removal is primarily

CONCLUSIONS AND RECOMMENDATIONS
The modelling exercise has demonstrated that suitable removal mechanisms can be integrally used to model a combined PoU system to predict the overall effluent bacterial quality. This kind of modelling can be used to optimize system design by allowing the engineer to systematically vary design parameters until the desired system effectiveness is attained. This research has also indicated that each barrier or treatment stage contributes to the overall E. coli removal.
Therefore, the bacterial load on the SCCGM (which is the main disinfection stage) can be significantly reduced by optimizing all components of the multi-barrier (combined) system, especially the pre-filtration stage. Some reasons for differences between predicted E. coli inactivation and actual inactivation by models such as the Chick's and It is recommended that future research should keep the obtained optimal flow constant then model the breakthrough of E. coli for several runs, ensuring water is passed in triplicate for each run. Furthermore, modelling of data obtained from field testing to assess possible applicability of the mathematical models on field data is proposed.
Also, concurrent modelling of E. coli and turbidity is proposed, since performance of filter systems is usually monitored by measuring effluent turbidity (MWH ).
Additionally, since the proposed multi-barrier water treatment design is scalable such that the capacity is flexible and can be increased to serve more consumers, modelling the effect of scalability is proposed. Long term experimentation is also recommended, and may help in calibrating model parameters to achieve the best fit between the modelled and measured values. Quantification of measured influent and effluent E. coli counts using particle counting techniques is also recommended. This may help characterize the modelled microbial particle diameter (d p ) better.