Mathematical Modeling of a Domestic Wastewater Treatment System Combining a Septic Tank, an Up Flow Anaerobic Filter, and a Constructed Wetland 78 American Academic Scientific Research Journal for Engineering, Technology, and Sciences ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 https://asrjetsjournal.org/index.php/American_Scientific_Journal/index Mathematical Modelling of a Composite Granular Filter of Effluent at Shirere Wastewater Treatment Plant in Kakamega County, Kenya Otenyo Makonjio Philipa*, Masibayi Namusasi Edwardb, K’Owino Owino Isaacc, Samuel Soita Chinad a,b,c,dMasinde Muliro University of Science and Technology, Department of Disaster Preparedness and Engineering Management, P.O. Box 190-50100 Kakamega, Kenya. aEmail: otenyo.philip@gmail.com bEmail: enamusasi@mmust.ac.ke cEmail: ikowino@mmust.ac.ke dEmail: schina@mmust.ac.ke Abstract Insufficient technology for municipal wastewater treatment compromises the quality of effluent discharged into water bodies, elevating the risk of waterborne diseases (e.g., cholera, dysentery, typhoid). Previous research has associated the absence of clean water and sanitation with health issues such as skin problems, eye infections, and diarrhea among community members. Furthermore, studies indicate the proliferation of algae in the Shirere wastewater oxidation ponds, suggesting the presence of toxic cyanobacteria. Therefore, this study aimed to develop a mathematical model representing five critical parameters: COD, BOD, TSS, Phosphates, and Nitrates. Effluent from Shirere WWTP were collected for microbial quality analysis at MMUST and KACUWASCO laboratories. Data analysis involved, regression and correlation, and integration of wastewater mass balance equation using R-Programming and Fourth Order Runga Kutta (RK) method. The research employed purposeful sampling strategy, with a sample size of 8 of wastewater. The study followed an experimental design. Specifically, for the first season of March – May 2021 at 200mm filtration depth were carried out at effluent flow rate of 0.0032π‘š3/𝑠 and volume, 0.234 π‘š3. the model arrived at was 𝐢 = 𝐢𝑖𝑛(1 βˆ’ π‘’βˆ’π‘‘(𝑄 𝑉⁄ ) . The model results showed minimal variation from the measured values. ------------------------------------------------------------------------ Received: 10/4/2023 Accepted: 11/8/2023 Published: 11/18/2023 ------------------------------------------------------------------------ * Corresponding author. The first season measured COD as 0.236kg/m3 and model gave 0.2174kg/m3. The model can be used in prediction of parameter concentrations at any given time. The findings of this research will inform wastewater management policies and contribute to the development of sustainable wastewater treatment technologies. https://asrjetsjournal.org/index.php/American_Scientific_Journal/index American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 79 Keywords: wastewater treatment; modelling; mass balance; composite granular filter; filter depth, Hydraulic retention time. 1. Introduction The importance of water quality cannot be overstated in relation to social and economic development, environmental sustainability, and human health. Nevertheless, with the ongoing expansion of the global population, the accessibility of sufficient and uncontaminated water resources for the entire population is progressively diminishing. The implementation of efficient wastewater treatment methods has the capacity to safeguard our ecosystems, while concurrently offering significant resources, such as fertilizers, [1,2,3]. Presently, a staggering 1.8 billion individuals across the globe are subjected to the use of water that is tainted, so placing them at risk of contracting waterborne illnesses, including but not limited to cholera, dysentery, typhoid, and polio, [4]. Numerous urban areas exhibit deficiencies in the necessary infrastructure and resources essential for the effective and sustainable management of wastewater, [5]. The phenomenon of water contamination experienced a significant deterioration in multiple watersheds located in Africa, Asia, and South America over the 1990s, as revealed by author, [6]. According to [7], it is anticipated that nations with low- and middle-income, notably in Africa, will see the most significant rises in pollution exposure as a result of their dense population. The issue of cyanobacterial contamination, which is associated with the production of toxins that pose a threat to the health of both humans and animals, is a matter of significant importance. The potential of zinc peroxide in the removal of red dye for water purification in many sectors has been explained by author, [8]. Nevertheless, this approach exhibits several constraints, such as its dependence on a predetermined temperature and UV radiation, alongside the requirement for intricate experimental configurations. According to [9], the existing methods of water and wastewater treatment in numerous African nations are insufficient. The primary methods utilized for wastewater treatment encompass onsite treatment, offsite treatment, conventional treatment, and stabilization ponds. Persistent challenges, such as inadequate infrastructure and inadequate operation maintenance, continue to hinder optimal performance. In the city of Addis Ababa, located in Ethiopia, the Kaliti treatment facility is seen to cater to a far less population than originally anticipated, primarily as a result of restricted sewer collection capabilities. The breakdown of pump stations in Kisumu, Kenya has led to the discharge of sewage into Lake Victoria. Recent research has conducted evaluations on the quality of water in many geographical areas. In the study of [10], conducted an investigation of the river Molo watershed in Kenya. Their findings revealed that the pH, temperature, fluorides, and sodium levels in the river exceeded the allowed limits set by the World Health Organization (WHO). The author [11], conducted an assessment of the potability of groundwater in Langata Sub County, Nairobi-Kenya, employing a Water Quality Index. The findings of their investigation exhibited encouraging implications for the effective management of groundwater resources. Inadequate water quality can give rise to a variety of health problems, encompassing a broad spectrum of water-related disorders. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 80 Moreover, the proliferation of cyanobacteria in aquatic environments presents a potential hazard to both human populations and wildlife. This issue is notably widespread in densely populated urban regions characterized by inadequate waste management procedures. Numerous metropolitan regions in Kenya encounter challenges pertaining to the management of wastewater and solid waste, resulting in the contamination of water resources such as the Athi River, [12]. The provision of water in Kakamega Town and its surrounding areas is predominantly dependent on the River Isiukhu and the Tindinyo gravity water system. Nevertheless, the existing water output is insufficient to meet the prevailing demand, hence necessitating the exploration of alternate sources such as shallow wells and rainfall collection. The field of solid waste management encounters various obstacles, including the prevalent practice of indiscriminate dumping and the resulting contamination of aquatic bodies. The absence of adequate sanitary infrastructure in Kakamega poses a significant health hazard to its inhabitants, as it increases their vulnerability to various diseases including as amoebic infections, bilharzia, typhoid fever, malaria, dysentery, and cholera, [13]. The Kenya Water and Sanitation Civil Societies Network (KEWSNET) conducted investigations and found that defective wastewater treatment plants are a notable source of pollution in major towns, such as Kakamega, [14]. The primary objective of this study is to investigate the problem of wastewater quality at the Shirere Wastewater Treatment Plant. The proposed approach involves the incorporation of a granular composite filter at the exit of maturation pond, with the intention of improving the quality of effluent discharged into the river Isiukhu. In the foreseeable future prediction of effluents discharged from the effluent filter is to be found by the mathematical model. Several mathematical models were applied for the prediction including mass balance model. Considerable attention has been devoted to investigating the phenomenon of flooding in bigger rivers. In summary, this research study examines the urgent concerns surrounding water quality, wastewater treatment, and public health in Kakamega Town and other urban settings. The primary objective is to develop mathematical model for prediction of several water parameters after installation of the filter. Efforts have been made to improve the quality of wastewater discharged into water bodies. Examples include a study by, [15] in which a vertical orientation of effluent flowing in the reactors from top to bottom. [16], in a vertical composite filter was using biochar and sand as filter media, flow of effluent was from top to bottom with high hydraulic transient which was a disadvantage. According to [17], revealed that, another vertical column filters composed of sand and pumice was used for removal of phosphates from top to bottom. However, none of these have been applied to improve wastewater discharged into River Isiukhu from Shirere Wastewater Treatment Plant. This study adopted a horizontal -flow based reactor filled with composite filter at exit maturation pond in Shirere Wastewater Treatment Plant. The filter materials consisted of granular sand and pumice stone at depths of 200mm, 400mm and 600mm. The effluent entered in a lateral manner by gravity through perforations in the front screen. It was controlled by the principle of differentials in hydraulic heads of effluents before and after filtration. It is believed waste materials get adsorbed onto the composite materials as it passes through the filter. 2. Materials and Methods 2.1. Study Area American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 81 The study was conducted at Shirere wastewater treatment plant consisting of a Composite Filter, S1 sampling site and S2 sampling site. The catchment area for Shirere WWTP, Shikoye stream and river Isiukhu is defined by longitude 34044’36.40’’E, 34045’8.65” E and latitude 0016’4.61” N, 0015’11.01” N. The location of the treatment plant is longitude 34044’55.85’E, 34044’53.93E and latitudes 0015’58.76” N, 0015’55.79” N. The wastewater from the plant is discharged into River Isiukhu via Shikoye stream. Figure 1 below gives the locations of the sampled points. Figure 1: Schematic Diagram of Shirere Wastewater Treatment Plant 2.2. Study Population Screen Influent Flow Influent From Septic Tank Effluent Discharged to Shikoye Stream Primary Facultative Pond ο‚· Aerobic Sediment ation ο‚· Secondary Facultative Pond Maturation Pond American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 82 This study used points S1 and S2 for analysis. 2.3. Research design This study employed experimental research design. The scientific analysis involved sample collection, preparation and laboratory analysis to determine among others concentration of COD, BOD, phosphates, total suspended solids (TSS) and nitrates while purposeful sampling technique was applied for data collection. Table 1: Research design for the study Specific objectives Approach Measurable Indicator Research Design Results To develop a mathematical model for wastewater treatment plant for managing the quality of the Isiukhu River Measure COD, BOD Nitrate, Phosphates and TSS. COD, BOD, Nitrates and Phosphates. Correlation and experimental designs. Tables and Graphical representation 2.4 Sampling Strategy Secondary data on effluent discharged and levels of contaminants in drinking water were obtained from Water Services Regulatory Board (WASREB). The sampling strategy for the scientific phase is shown in Table 2. Table 2: Sampling strategies for the Scientific Phase Study unit Sampling methods Sample size Wastewater purposeful 8, 4 from each point of 2 sampling sites 2.5 Data Collection The triplicates collected at sites S1-S2 were included for control sample conditions using labelled 500 millilitres sampling bottles and kept in ice boxes. In summary, samples were collected at the outlet of the maturation pond before filtration (S1). The above was repeated with experimental sample conditions collected after the introduction of sand-pumice composite filter within the exit chamber that created a three section of S1, reactor and S2. New Samples were therefore collected in S1, and S2. All samples were taken to MMUST laboratory for further analysis. This was done during the dry season, wet season and short rain season. 2.5.1 Particle size determination American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 83 Several particle sizes were obtained after passing ground pumice stones through various sieves. Particle size preparation was undertaken by mechanical sieving which is more efficient, [18]. In the British standard, sieving test procedure, a Bs 410 standard sieves are used. The sieve numbers are from 4 to 270 and corresponding sieves sizes ranges from 0.053 mm to 4.75mm, [19]. In this study the available sizes at MMUST Civil engineering laboratories had sieves sizes 0.6mm, 0.9mm, 1.18, 2.75mm and 3.25mm. 2.5.1.1 Optimal particle size determination To obtain optimal particles sizes for filter material laboratory experiments involving determination of maximum wavelength (Lambda mark), and development of standard calibration curve were carried out. The optimum absorption wavelength Lambda mark for phosphate ions was determined in the range between 500nm to 950nm, [20]. A stock solution of 0.01mole of potassium dihydrogen phosphate was prepared by dissolving 1.36g of the salt into 500millilitres of distilled water and whose absorbance was measured in UV-vis spectrophotometer by varying the wavelength from 500nm to 950nm. Three trials of absorbance were carried for every wavelength in the range given and mean absorbance recorded. Using recorded results, the parameters of Gaussian function, 𝐴 = π΄π‘šπ‘Žπ‘₯exp [ πœ†βˆ’π‘š 𝑠 ]2 were determined by non-linear curve fitting in MATLAB and their optimal values were found as follows:π΄π‘šπ‘Žπ‘₯ = 0.65, π‘š = 752.02π‘›π‘š, and 𝑠 = 146.24π‘›π‘š. Figure 2: Plot of wavelength versus absorbance for the determination of optimum wavelength From this graph, at maximum absorbance, using differential calculus, the lambda mark was found to be π‘š = 752.02π‘›π‘š. Author, [20] conducted research using the same ratios and materials and revealed found lambda mark as 713π‘›π‘š and [21], at πœ† = 715 π‘›π‘š which compares well with our results. 2.5.1.2 Standard Calibration curve Obtained Lambda wavelength ( πœ† = 752) was fixed on UV-VIS spectrophotometer for measurements of ] 24.146 )02.752( exp[65.0 2 2ο€­ ο€­ο€½  A 9938.02 ο€½R π‘Šπ‘Žπ‘£π‘’π‘™π‘’π‘›π‘”π‘‘β„Ž, πœ†(π‘›π‘š) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 84 absorbance of stock solution having concentrations of 5ppM, 10ppM, 15ppM, 20ppM , 25ppM, and 30ppM and recorded. Plotting stock concentrations against absorbance yielded standard calibration curve as shown in Figure 3. Figure 3: Calibration curve showing a plot of concentration in ppm against absorbance of 0.01M potassium dihydrogen phosphate in distilled water Absorbance was examined before and after filtering the stock solution through three cylinders with composite materials, each with a 50% mixture of 1.18 mm pumice and 1.18 mm sand, 1.18 mm sand and 0.9 mm pumice, and 1.18 mm sand and 0.6 mm pumice filtering material. The ideal composite filter was chosen using absorbance variation. The combination of 50% sand (1.18 mm) and 50% pumice (0.6 mm) at 0.248 absorbance variance was determined to have the best absorbance variance. 2.6 Adsorption time Figure 4 shows the variation in phosphorous absorbance against incubation time in batch experimental conditions. In this case, waste water collected from Site S1, was used in batch experiment at MMUST chemistry laboratories. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 85 Figure 4: Plot of time versus absorbance variance (%) The phosphate removal efficiencies were 53%, 58%, 66%, 72%, 74% and 77% for contact time of a half an hour, 1, 2, 4, 8, 16, 32 and 64 hours respectively. The rate of phosphates removed under these conditions increased sharply up to 8 hours and gradually reached equilibrium after 16 hours reaction time. Thus, the equilibrium 77% adsorption of phosphates by the composite filter reached was attained after 16 hours beyond which minimum was observed. Inset is an adsorption model for the same composite filter developed for the same experimental conditions. This still showed 77% asymptotically success for phosphate removal. The obtained results showed that the adsorption forces followed a decaying experimental equation of 𝑦 = 75.53 exp(0.0003745π‘₯) βˆ’ 29.45exp (βˆ’0.53322π‘₯) a correlation coefficient of 0.994. This 16 hour was used by the researcher as minimal time after installation of composite filter reactor and material to the samples from S1 – S2. 2.7 Reactor design, and Material Installation The main element of this study setting is the wastewater filtering apparatus. It is necessary for carrying out filtration studies and is specially made to fit the maturation pond's exit S1, [22]. According to Figure 5a, it was made to match the maturation pond's S1 exit. It measured 1170 mm in length, 1000 mm in breadth, and 600 mm in height to be fabricated of stainless steel. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 86 Figure 5a: Design of Reactor/wastewater filtration instrument Fabricated water-tight reactor with filter was installed into maturation pond outlet to fit its length and depth. This created sample collection zones along the width labeled S1 and S2 as previously described. Wastewaters were allowed to pass through it for 24 hours and samples were collected for analysis at points S1-S2. This was carried out to each at every 200mm, 400mm and 600mm depth of the composite filter into the maturation pond as shown in figure 5b below. Figure 5b: Installed Composite granular filter in the reactor with horizontal inflow and outflow of effluent 2.8. Water Quality Sampling Fabricated water-tight reactor was installed into maturation pond outlet to fit its length and depth. This created sample collection zones along the width labeled S1 and S2 as previously described. Composite materials were American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 87 filled into it at heights of 200 mm, 400 mm and 600 mm. Wastewaters were allowed to pass through it overnight for 24 hours and samples collected for analysis at points S1 and S2. This was carried out to each at every 200mm, 400mm and 600mm depth of the composite filter at the maturation pond exit. Triplicate samples were collected during the seasons of March to May, June to August and September to November 2021. The samples were collected in labelled 500 milliliters bottles sterilized by HNO2 acid and stored under controlled container conditions before being transported to the laboratories. Three bottles per site were collected to enable sufficient analysis in the MMUST chemical laboratories, [23]. At the MMUST chemistry laboratory the field samples were filtered through 0.4-Β΅m pore membrane filters and kept at 40C until analysis. An aliquot of 50 milliliters of these used were digested with 20 milliliters of HCl acid at 800C until the solution became transparent, [24]. All samples were analyzed to determine biological oxygen demand (BOD), chemical oxygen demand (COD), nitrates (𝑁𝑂3), phosphate (𝑃3𝑂), and total suspended solids (TSS). 2.9. Mass Balance Model 2.9.1 Model Assumptions The mass-balance analysis is the most basic method for determining the changes that occur when a reaction occurs in a container (reactor) or in a defined area of a body of liquid, [25]. Figure 6 depicts the mass balance for a single reactor in this research. Figure 6: The system representation of WWTP using a composite filter American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 88 The system is displayed in Figure 6 where 𝑄𝑖𝑛 and π‘„π‘œπ‘’π‘‘ are the volumetric flow rate into and out of the reactor. inc = concentration of effluent into the reactor, and c = concentration of effluent out of the reactor. To apply a mass-balance analysis to the liquid contents of the reactor shown in Figure 6 above, the following assumptions were made: i). the volumetric flow rate of effluent out of the reactor is equal to the volumetric flow rate of effluent into the reactor, 𝑄𝑖𝑛 = π‘„π‘œπ‘’π‘‘ = 𝑄, ii). the liquid within the reactor is not subject to evaporation (isothermal conditions), and iii). No reactions in the process of filtration. 2.9.2 Model Description For the stated assumptions, the materials mass balance can be formulated as follows: )( CC V Q dt dC in ο€­ο€½ (1) If only the steady-state effluent concentration is sought, Eq. (4) can be simplified by noting that the rate accumulation is zero under steady-state conditions (dc/dt= 0), [26]. From Eq. (1), steady state is given as; inCC ο€½ (2) Eq. (1) is a first order ordinary differential equation with variables separable which can be easily integrated as follows. Integrating between the limits of 0 and 𝐢 and 0 and 𝑑, and solving yields: C=Cin (1-π‘’βˆ’π‘‘π‘‘0) (3) where 𝑑0 = 𝑄 𝑉⁄ is ponding time. In this study, the parameters 𝐢𝑖𝑛 and 𝐢 were replaced with inBOD , inCOD , in NO3 , in PO3 , inTSS and BOD , COD , 3NO , 3PO , and TSS respectively. MATLAB R2021a was used to plot the concentration curve for each water quality parameter. 2.10 Limitation of the Study This study experienced inadequate sealing of the effluent on the sides of the reactor in the outlet manhole of maturation pond. This means that, the effluent was not fully filtered through the reactor. The limitation was addressed by incorporating more silica material on its external sides to reduce leakage. 3. Results and discussions This section shows the results of the study and their discussion. It also includes establishment of a mathematical mass balance model of the filter performance and seasonal variability of pollutant removal from effluents. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 89 3.1. Mathematical Mass Balance Model To establish the mass balance model, data analysis was done by use of descriptive statistics of the parameters, standard deviations, means, frequencies, after which inferences were drawn from the analyses The mass balance model was given as: )( )( )( )( )( 33 3 33 3 TSSTSS V Q dt dTSS NONO V Q dt dNO POPO V Q dt dPO CODCOD V Q dt dCOD BODBOD V Q dt dBOD in in in in in ο€­ο€½ ο€­ο€½ ο€­ο€½ ο€­ο€½ ο€­ο€½ (4) These equations were integrated using the fourth-order Runga Kutta (RK) method. The initial values of 𝐢𝑖𝑛(𝐡𝑂𝐷𝑖𝑛 , 𝐢𝑂𝐷𝑖𝑛 , 𝑇𝑆𝑆𝑖𝑛 , 𝑃𝑂3𝑖𝑛 , 𝑁𝑂3𝑖𝑛) , 𝐢 ( 𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑃𝑂3, 𝑁𝑂3 ,, and 𝑑(π‘‘π‘–π‘šπ‘’) for the method are given in Tables. The analytical solution for the same system of equations is given by Eq. 5 as shown below. 𝐢 = 𝐢𝑖𝑛(1 βˆ’ π‘’βˆ’π‘‘(𝑄 𝑉⁄ )) Which translates to specific model shown below; )1( )1( )1( )1( )1( )/( 33 )/( )/( 33 )/( )/( tVQ in tVQ in tVQ in tVQ in tVQ in eNONO eTSSTSS ePOPO eCODCOD eBODBOD ο€­ ο€­ ο€­ ο€­ ο€­ ο€­ο€½ ο€­ο€½ ο€­ο€½ ο€­ο€½ ο€­ο€½ (5) The parameter 𝑄 (flow rate) was determined using Arcostic Doppler Velocimeter (ADV), while volume, V, was obtained by multiplying the depth of the filter and the reactor area. Using the values obtained by RK, non-linear least square method was applied to estimate the parameters of the above system of equations. The mass balance equations for each season, flow rate, and depth were integrated via the use of the fourth-order Runga Kutta (RK) method for systems of equations given by Eq. (5). The results and discussions are given in the subsections 3.1 – 3.9. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 90 3.1. 1 March – May 2021 at 200mm filter depth March – May 2021 at 200mm filtration depth were carried out at effluent flow rate of 0.0032π‘š3/𝑠 and volume, 0.234 π‘š3. Tables 3 provided 𝐢𝑖𝑛 and 𝐢 as starting values for numerical integration of Eq. (4, Table 3: Concentrations of 𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑃𝑂3 and 𝑁𝑂3 filtration for March – May season at 200mm and 0.0032π‘š3/𝑠 Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢𝑖𝑛 0.0241 0.0199 0.3002 0.2813 0.3252 After 𝐢 0.0220 0.0161 0.1909 0.2726 0.2913 The results of the integrations are shown in Figure 7. The corresponding equations of the graphs of Figure 7 are given by table 3 Figure 7: Concentrations of BOD , COD , TSS , 3NO and 3PO versus time with filter at depth of mm200 , 3234.0 m of volume, and input discharge of sm /0032.0 3 for March-May season American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 91 Table 4: Result Summary, season 1, at 200mm Season Depth (mm) Parameters Cin 3/ mKg Q(π’ŽπŸ‘/𝒔) V (π’ŽπŸ‘) V/Q= (Detention t/s) Q/V(t0) Concentratio n (C) at t0 (π‘²π’ˆ/π’ŽπŸ‘) March- may 200 COD 0.3002 0.0032 0.234 73 0.0137 0.1891 BOD 0.2813 0.0032 0.234 73 0.0137 0.1772 TSS 0.3252 0.0032 0.234 73 0.0137 0.2049 NO3 0.0199 0.0032 0.234 73 0.0137 0.0125 PO3 0.0241 0.0032 0.234 73 0.0137 0.0152 From eq. 5, the concentration of each of the parameters increased until it reached steady-state. The steady-states of the graphs of Figure 7 given by Eqs. 5 was obtained by differentiating these equations, equating to zero, and the resulting concentrations was determined as shown in table 4. From the table, the model was predicting well the COD concentration at S2 sampling point which had very minimal variance. Measured COD was 0.1909 3/ mKg and the model predicted COD was 0.1891 3/ mKg . The Measured Phosphates was, 0.0220 kg/m3 while the model predicted phosphates was 0.0152 kg/m3 this indicates a minimal variance of 0.0078 kg/m3 implying that the model is accurate. 3.1.2March – May 2021 at 400mm filtration depth March – May 2021 400mm filtration depth were determined at effluent flow rate of 0.0036 π‘š3/𝑠 and 0.468 π‘š3 of volume. The concentrations, 𝐢𝑖𝑛 before and 𝐢 after for BOD , COD , TSS , 3NO and 3PO represented in Table 5. Table 5: Concentrations of 𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑃𝑂3 and 𝑁𝑂3 filtration for Season March – May at 400mm depth Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢in 0.0291 0.3278 0.3559 0.3002 0.3654 After 𝐢 0.0255 0.0272 0.2311 0.2827 0.3305 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 92 These provided the initial values for the Runga Kutta (RK) method. The integration of the system of Eq. (4) using fourth-order Runga Kutta (RK) method for systems of equations gave the graphs shown Figure 8. Figure 8: Concentrations of 5BOD , COD , TSS , 3NO and 3PO versus time with filter at depth of mm400 , 3468.0 m of volume, and input discharge of sm /0036.0 3 for March-May season Table 6: Result Summary for season 1, at 400mm Season Depth (mm) Parameters Cin (π‘²π’ˆ/π’ŽπŸ‘) Q=(π’ŽπŸ‘/𝒔 V(π’ŽπŸ‘) V/Q= (Detention time, sec) Q/V(t0) Concentrat ion (C) at t0(π‘²π’ˆ/π’ŽπŸ‘) March- may 400 COD 0.3559 0.0036 0.468 130 0.00769 0.2242 BOD 0.3002 0.0036 0.468 130 0.00769 0.1891 TSS 0.3654 0.0036 0.468 130 0.00769 0.2302 NO3 0.3278 0.0036 0.468 130 0.00769 0.2065 PO3 0.0291 0.0036 0.468 130 0.00769 0.0183 From Eqs. 5, the concentration of each of BOD ,COD , TSS , 3NO and 3PO increased until it reached steady-state. For steady-state, concentrations was as follows: 𝐡𝑂𝐷 = 0.300𝐾𝑔/π‘š3 , 𝐢𝑂𝐷 = 0.3559𝐾𝑔/ π‘š3 , 𝑇𝑆𝑆 = 0.3654𝐾𝑔/π‘š3, 𝑃𝑂3 = 0.029𝐾𝑔/π‘š3 , 𝑁𝑂3 = 0.0328𝐾𝑔/π‘š3 and detention time, 𝑉 𝑄⁄ 130ο€½ seconds. From table 6 the model was predicting well the COD concentration at S2 which had very minimal variance. Measured COD was 0.2311 3/ mKg and the model COD was 0.2242 3/ mKg . In addition, the American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 93 measured PO3 was 0.0255 kg/m3, and the model predicted PO3 0.0183 kg/m3. This also displays a minimum variance of 0.0072 kg/m3, which implies that the model is accurate in predicting the parameters, [27]. 3.1.3 March – May 2021 at 600 mm filter depth March – May 2021 at 600mm filter depth corresponded to 0.702π‘š3 of the filter and effluent flow rate of 0.0045π‘š3/𝑠 . The values of the concentration of BOD, COD, TSS, PO3 and NO3 before, 𝐢𝑖𝑛 and after, 𝐢 are given Table 7. Table 7: Concentrations of 𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑃𝑂3 and 𝑁𝑂3 filtration for March – May season at 600mm Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢𝑖𝑛 0.0278 0.0319 0.2967 0.3220 0.3416 After 𝐢 0.0221 26.783 0.1694 0.2853 0.3010 The 𝐢𝑖𝑛 , 𝐢, and 𝑑 = 0 were the starting initial values of the system of differential equations given by Eq. (4). The integration of the system of Eq. (4) using fourth-order Runga Kutta (RK) method for systems of equations and the results are depicted in Figure 9. Figure 9: Concentrations of 5BOD , COD , TSS , 3NO and 3PO versus time with filter at depth of mm600 , 3702.0 m of volume, and input discharge of sm /0045.0 3 for March-May season American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 94 Table 8: Result Summary, season 1, at 600mm Season Depth (mm) Parameter s Cin(π‘²π’ˆ/π’ŽπŸ‘) Q=(π’ŽπŸ‘/𝒔 V(π’ŽπŸ‘) V/Q=(De tention time, sec) Q/V(t0) Concentr ation (C) at t0 (π‘²π’ˆ/ π’ŽπŸ‘) March- may 600 COD 0.2967 0.0045 0.7020 156 0.0064 0.1869 BOD 0.3220 0.0045 0.7020 156 0.0064 0.2029 TSS 0.3416 0.0045 0.7020 156 0.0064 0.2152 NO3 0.0319 0.0045 0.7020 156 0.0064 0.0201 PO3 0.0278 0.0045 0.7020 156 0.0064 0.0175 From Eqs. 5, the concentration of each of the BOD, COD, TSS, PO3 and NO3 increased until it reached maximum. For steady-state, concentrations was as follows: 𝐡𝑂𝐷 = 0.3220𝐾𝑔/π‘š3 , 𝐢𝑂𝐷 = 0.2967𝐾𝑔/π‘š3 , 𝑇𝑆𝑆 = 0.3416𝐾𝑔/π‘š3, 𝑃𝑂3 = 0.0280𝐾𝑔/π‘š3, 𝑁𝑂3 = 0.0319𝐾𝑔/π‘š3 and detention time, 𝑉 𝑄⁄ 156ο€½ seconds. The model in table 8 was predicting accurately the COD concentration at S2 which had very minimal variance. Measured COD was 0.1694 3/ mKg and the model COD was 0.1869 3/ mKg . The measured PO3 was 0.0221 kg/m3 while the model predicted 0.0175 kg/m3, which has a negligible variance. The for three different depths, 200mm, 400mm, 600mm corresponding to volume: 3468.0 m , 3234.0 m , 3702.0 m , gave the following detention times; 1.73 seconds, 130 seconds and 156 seconds. This is an indication that the detention time increases with the volume of the filter. 3.1.4 June – August at 200mm filter depth June – August season effluent flow rate of 0.0039 π‘š3/𝑠 and volume, 0.234 π‘š3 . The concentrations of concentrations of 𝐢𝑖𝑛 and 𝐢 for BOD , COD , TSS , 3NO and 3PO before, and after are represented in Table 9. Table 9: Concentrations of BOD , COD , TSS , 3NO and 3PO for March – May season at 200mm depth Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢𝑖𝑛 0.0221 0.0179 0.2702 0.2813 0.3002 After 𝐢 0.0200 0.0141 0.1609 0.2526 0.2613 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 95 The values in the Table 10 were used as the starting initial values of RK method for Eq. (4,. Integration of the system of differential equation, Eq. (4, yielded Figure 10: Concentrations of 5BOD , COD , TSS , 3NO and 3PO versus time with filter at depth of mm200 , 3234.0 m of volume, and input discharge of sm /0039.0 3 for June-August season Table 10: Result Summary, season 2, at 200mm Season Depth (mm) Parameter s Cin (π‘²π’ˆ/π’ŽπŸ‘) Q=(π’ŽπŸ‘/𝒔 V(π’ŽπŸ‘) V/Q=(De tention t/s) Q/V(t0) Concentr ation (C) at t0 (π‘²π’ˆ/ π’ŽπŸ‘) June- August 200 COD 0.2702 0.0039 0.234 60 0.0167 0.1702 BOD 0.2813 0.0039 0.234 60 0.0167 0.1772 TSS 0.3002 0.0039 0.234 60 0.0167 0.1891 NO3 0.0179 0.0039 0.234 60 0.0167 0.0113 PO3 0.0221 0.0039 0.234 60 0.0167 0.0139 From Eqs. 5, the concentration of each of the variables (𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑃𝑂3, 𝑁𝑂3) increased until it reached maximum. Differentiating these equations and equating zero, the resulting concentrations were 𝐡𝑂𝐷 = 0.2813𝐾𝑔/π‘š3 , 𝐢𝑂𝐷 = 0.2702𝐾𝑔/π‘š3 , 𝑇𝑆𝑆 = 0.3002𝐾𝑔/π‘š3, 𝑃𝑂3 = 0.0221𝐾𝑔/π‘š3 , 𝑁𝑂3 = 0.0179𝐾𝑔/ American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 96 π‘š3 and detention time, 𝑉 𝑄⁄ 60ο€½ seconds. The model prediction in table 10 was accurately performing for the COD concentration at S2 which had very minimal variance. Measured COD was 0.1609 3/ mKg and the model COD was 0.1702 3/ mKg . Additionally, the measured PO3 was 0.0020 kg/m3, while the prediction of the model was 0.0139 kg/m3. 3.1.5 June – August at 400mm filter depth The experiment was carried out at effluent flow rate of 0.0042π‘š3/𝑠 and 0.468π‘š3. Table 11: Concentration values before and after filtration for June – August season at 400mm and 0.0042π‘š3/𝑠 Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢in 0.0212 0.0228 0.3209 0.2702 0.3254 After 𝐢 0.0204 0.0173 0.2111 0.2427 0.3155 Table 11 provided the initial concentration values for system of differential equations. The integration of the system of Eq. (4) using fourth-order Runga Kutta (RK) method for systems of equations and the results are as follows. Figure 11: Concentrations of 𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑁𝑂3, and 𝑃𝑂3 versus time with filter at depth of 400 π‘šπ‘š, 0.468 π‘š3of volume, and input discharge of 0.0042 π‘š3/𝑠 for June-August season American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 97 Table 12: Result Summary, season 2, at 400mm Season Depth (mm) Parameters Cin (𝐾𝑔/π‘š3, Q= (π‘š3/𝑠, V (π‘š3) V/Q= (Detenti on time, sec) Q/V(t0) Concent ration (C) at t0 (𝐾𝑔/ π‘š3, June- August 400 COD 0.3209 0.0042 0.468 111 0.0090 0.2021 BOD 0.2702 0.0042 0.468 111 0.0090 0.1702 TSS 0.3254 0.0042 0.468 111 0.0090 0.2050 NO3 0.0218 0.0042 0.468 111 0.0090 0.0137 PO3 0.0212 0.0042 0.468 111 0.0090 0.0134 From Eqs. 5, the concentration of each of the variables (𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑃𝑂3, 𝑁𝑂3) increased until it reached maximum. Differentiating these equations and equating zero, the resulting concentrations were 𝐡𝑂𝐷 = 0.2702𝐾𝑔/π‘š3 ,𝐢𝑂𝐷 = 0.321𝐾𝑔/π‘š3 ,𝑇𝑆𝑆 = 0.3254𝐾𝑔/π‘š3 , 𝑃𝑂3 = 0.0212𝐾𝑔/π‘š3 , 𝑁𝑂3 = 0.02178𝐾𝑔/π‘š3 and detention time, 𝑉 𝑄⁄ 111ο€½ seconds. The model in table 12 was predicting accurately the COD concentration at S2 which had very minimal variance. Measured COD was 0.2111 3/ mKg and the model COD was 0.2021 3/ mKg . Also, the measured PO3, was 0.0204 and the predicted model was 0.0134 kg/m3 , which is a minimal variance. 3.1.6 June – August season at the filter depth of 600mm June – August filtration experiments were determined using effluent flow rate of 0.0045π‘š3/𝑠 , and volume 3702.0 m Table 13: Concentrations of BOD , COD , TSS , 3NO and 3PO before and after filtration for June – August season at 600mm and 0.0045m3/s Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢𝑖𝑛 0.0256 0.0318 0.2567 0.3313 0.3812 After C 0.0221 0.0228 0.1619 0.2856 0.3413 Table 13 provided the initial parameter values and the state constants for following system of differential equations. The integration of the system of Eq. (4) results to the following Eqs. 4.2 and their plots are given in Figure 12. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 98 Figure 12: Concentrations of 5BOD , COD , TSS , 3NO and 3PO versus time with filter at depth of mm600 , 3702.0 m of volume, and input discharge of sm /0045.0 3 for June -August season Table 14: Result Summary, season 2, at 600mm Season Depth (mm) Parameters Cin Q=(π’ŽπŸ‘/𝒔 V(π’ŽπŸ‘) V/Q=(Detentio n time, sec) Q/V(t0) Concentrati on (C) at t0 (π‘²π’ˆ/π’ŽπŸ‘) June- August 600 COD 0.2567 0.0045 0.702 156 0.0064 0.1617 BOD 0.3313 0.0045 0.702 156 0.0064 0.2087 TSS 0.3812 0.0045 0.702 156 0.0064 0.2402 NO3 0.0318 0.0045 0.702 156 0.0064 0.0200 PO3 0.0256 0.0045 0.702 156 0.0064 0.0161 From Eqs. 5, the concentration of each of the variables (𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑃𝑂3, 𝑁𝑂3) increased until it reached maximum. Differentiating these equations and equating zero, the resulting concentrations were 𝐡𝑂𝐷 = 0.3313𝐾𝑔/π‘š3 ,𝐢𝑂𝐷 = 0.2567𝐾𝑔/π‘š3 ,𝑇𝑆𝑆 = 0.3812𝐾𝑔/π‘š3 , 𝑃𝑂3 = 0.0256𝐾𝑔/π‘š3 , 𝑁𝑂3 = 0.0318𝐾𝑔/π‘š3 and detention time, 𝑉 𝑄⁄ 156ο€½ seconds. The model in table 14 was predicting accurately the COD concentration at S2 which had very minimal variance. Measured COD was 0.1619 3/ mKg and the model COD was 0.1617 3/ mKg . The measured PO3 was 0.021 kg/m3, while the prediction from the model was, 0.0161 kg/m3 showing a small variance. Therefore, for June – August season, the three different depths, 200mm, 400mm, 600mm corresponding to volume: 3468.0 m , 3234.0 m , 3702.0 m , gave the following detention times; 60 seconds, 111 seconds and American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 99 156 seconds. This is an indication that the detention time increases with the increase in volume of the filter. 3.1.7 September – November season at the filter depth of 200mm The experiment was performed at effluent flow rate, 0.0039π‘š3/𝑠 and 0.234 π‘š3 of volume of the filter. Table 15: Concentrations of BOD , COD , TSS , 3NO and 3PO before and after filtration for September – November season at 200mm and 0.0039π‘š3/𝑠 Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢𝑖𝑛 0.0313 0.0251 0.3450 0.3071 0.3356 After 𝐢 0.0290 0.0221 0.2360 0.2973 0.2984 Figure 14: Concentrations of BOD , COD , TSS , 3NO and 3PO versus time with filter at depth of mm200 , 3234.0 m of volume, and input discharge of sm /0039.0 3 for June-August season American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 100 Table 16: Result Summary, season 3, at 200mm Season Depth (mm) Parameters Cin (π‘²π’ˆ/π’ŽπŸ‘) Q=(π’ŽπŸ‘/𝒔 V(π’ŽπŸ‘) V/Q=(Dete ntion time, sec) Q/V(t0) Concentrat ion (C) at t0(π‘²π’ˆ/π’ŽπŸ‘) Sept-Nov 200 COD 0.3450 0.0039 0.234 60 0.0167 0.2174 BOD 0.3071 0.0039 0.234 60 0.0167 0.1935 TSS 0.3356 0.0039 0.234 60 0.0167 0.2114 NO3 0.0251 0.0039 0.234 60 0.0167 0.0158 PO3 0.0313 0.0039 0.234 60 0.0167 0.0197 From Eqs. 5, the concentration of each of the variables (𝐡𝑂𝐷, 𝐢𝑂𝐷, 𝑇𝑆𝑆, 𝑃𝑂3, 𝑁𝑂3) increased until it reached maximum. Differentiating these equations and equating zero, the resulting concentrations were 𝐡𝑂𝐷 = 0.307𝐾𝑔/π‘š3 , 𝐢𝑂𝐷 = 0.345𝐾𝑔/π‘š3 ,𝑇𝑆𝑆 = 0.3356𝐾𝑔/π‘š3 , 𝑃𝑂3 = 0.0313𝐾𝑔/π‘š3 , 𝑁𝑂3 = 0.0251𝐾𝑔/π‘š3 and detention time, 𝑉 𝑄⁄ 60ο€½ seconds. The model in table 16 was predicting accurately the COD concentration at S2 which had very minimal variance. Measured COD was 0.2360 3/ mKg and the model COD was 0.2174 3/ mKg . The prediction from the model showed PO3 of 0.0197 kg/m3 , while the measured PO3 was 0.0290 kg/m3 3.1.8 September – November season at the filter depth of 400mm September – November season at the filter depth of 400mm corresponded to 3468.0 m of volume and effluent flow rate of 0.0042π‘š3/𝑠. Table 17: Concentrations of BOD , COD , TSS , 3NO and 3PO before and after filtration for September – November season at 400mm and 0.0036π‘š3/𝑠 Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢in 0.0367 0.0461 0.2650 0.3010 0.3803 After 𝐢 0.0325 0.0385 0.1458 0.2902 0.3302 Table 17 provided the initial parameter values and the state constants for following system of differential equations. The integration of the system of Eq. (4) using fourth-order Runga Kutta (RK) method for systems of equations and the results are as follows. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 101 Figure 14: Concentrations of 5BOD ,COD , TSS , 3NO and 3PO versus time with filter at depth of mm400 , 3468.0 m of volume, and input discharge of sm /0036.0 3 for March-May season where Eqs. 5 are the equations for graphs shown in Figure 14. Table 18: Result Summary, season 3, at 400mm Season Depth (mm) Parameter s Cin (π‘²π’ˆ/π’ŽπŸ‘) Q=(π’ŽπŸ‘/𝒔 V(π’ŽπŸ‘) V/Q=(De tention time, sec) Q/V(t0) Concent ration (C) at t0 (π‘²π’ˆ/ π’ŽπŸ‘) Sept- Nov 400 COD 0.2650 0.0036 0.468 130 0.0077 0.1670 BOD 0.3010 0.0036 0.468 130 0.0077 0.1896 TSS 0.3803 0.0036 0.468 130 0.0077 0.2396 NO3 0.0461 0.0036 0.468 130 0.0077 0.0290 PO3 0.0367 0.0036 0.468 130 0.0077 0.0231 From Eqs. 5, the concentration of each of the parameters increased until it reached maximum. Differentiating these equations and equating zero, the resulting concentrations were 𝐡𝑂𝐷 = 0.300𝐾𝑔/π‘š3 , 𝐢𝑂𝐷 = 0.3559𝐾𝑔/π‘š3 , 𝑇𝑆𝑆 = 0.3654𝐾𝑔/π‘š3 , 𝑃𝑂3 = 0.029 , 𝑁𝑂3 = 0.0328𝐾𝑔/π‘š3 and detention time, 𝑉 𝑄⁄ 130ο€½ seconds. The model in table 18 was predicting accurately the COD concentration at S2 which had very minimal variance. Measured COD was 0.1458 3/ mKg and the model COD was 0.1670 3/ mKg . The measured TSS was 0.3302 kg/m3 and the prediction from the model was 0.2396 kg/m3 which is equally a minimal variance. Just as it was revealed by a similar study, [27], this implies that the model can perfectly be American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 102 used for prediction of COD, BOD, TSS, as well as other related parameters. 3.1.9 September – November 600mm filter depth September - November 600mm filter depth, corresponded filter volume of 0.702π‘š3 and effluent flow rate of 0.0045π‘š3/𝑠. Table 19 represents the concentration values of variables, 𝐢𝑖𝑛 and 𝐢. Table 19: Concentrations of BOD , COD , TSS , 3NO and 3PO before and after filtration for September – November season at 600mm Filtration Concentration (𝐾𝑔/π‘š3, 𝑃𝑂3 𝑁𝑂3 COD BOD TSS Before 𝐢𝑖𝑛 0.0222 0.0278 0.2834 0.3172 0.3692 After 𝐢 0.0183 0.0214 0.1820 0.2676 0.3286 Table 19 provided the initial concentration values for the system of differential equations, Eq. (4). The integration of the system of Eq. (4) resulted to Eqs. 5 and their plots are given in Figure 15. Figure 15 : Concentrations of 5BOD , COD , TSS , 3NO and 3PO versus time with filter at depth of mm600 , 3702.0 m of volume, and input discharge of sm /0045.0 3 for September - November season \ American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 103 Table 20: Result Summary, season 3, at 600mm Season Depth (mm) Parameter s Cin (π‘²π’ˆ/π’ŽπŸ‘) Q=(π’ŽπŸ‘/𝒔 V(π’ŽπŸ‘) V/Q=(De tention time, sec) Q/V(t0) Concentr ation (C) at t0 (π‘²π’ˆ/ π’ŽπŸ‘) Sept-Nov 600 COD 0.2834 0.0045 0.702 156 0.0064 0.1785 BOD 0.3172 0.0045 0.702 156 0.0064 0.1998 TSS 0.3692 0.0045 0.702 156 0.0064 0.2326 NO3 0.0278 0.0045 0.702 156 0.0064 0.0175 PO3 0.0222 0.0045 0.702 156 0.0064 0.0140 From Eqs. 5, the concentration of each of the variables increased until it reached maximum. Differentiating these equations and equating zero, the resulting concentrations were 𝐡𝑂𝐷 = 0.317𝐾𝑔/π‘š3 , 𝐢𝑂𝐷 = 0.283𝐾𝑔/ π‘š3 , 𝑇𝑆𝑆 = 0.369𝐾𝑔/π‘š3 , 𝑃𝑂3 = 0.022𝐾𝑔/π‘š3 , 𝑁𝑂3 = 0.0328𝐾𝑔/π‘š3 and detention time, 𝑉 𝑄⁄ 156ο€½ seconds. The model in table 20 was predicting accurately the COD concentration at S2 which had very minimal variance. Measured COD was 0.1820 3/ mKg and the model COD was 0.1785 3/ mKg . While the measured PO3 was 0.0183 kg/m3 and the model predicted PO3 of 0.0140 kg/m3 which implies that the model is more accurate since the variance is minimal. Also noted, for the September – November season, the three different depths, 200mm, 400mm, 600mm corresponding to volume: 3468.0 m , 3234.0 m , 3702.0 m , gave the following detention times; 60 seconds, 130 seconds and 156 seconds. This is an indication that the detention time increases with the volume of the filter. 4. Conclusion This study therefore concludes that, the model developed provides valuable information regarding performance of the composite filter and natural wetland downstream on wastewater treatment. Mass balance model can be used for a concentration prediction at any given time. This eventually reduces time and cost of other measurements. Recommendation of the Study The mathematical model for composite granular filter for managing the quality of effluent is recommended for prediction of any of the parameters by substituting the values for time. This will enable the design engineers to design the system that can still maintain the desired output but vary volumes and flow rates of the reactors American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) - Volume 96, No 1, 78 -106 104 Acknowledgements Authors are grateful to almighty God strength, grace and knowledge to undertake this study. 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