Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 186 STOCHASTIC MODELS OF SOME PROPERTIES OF WASTE WATER IN THE MAAMERA SEWAGE TREATMENT PLANT Dr. Nabaa Shakir Hadi nabaalshimmri@yahoo.com College of Engineering, University of Babylon, Babylon, Iraq Received 30 November 2014 Accepted 3 February 2015 ABSTRACT The records of influents concentration for both BOD5 and TSS of Maamera sewage treatment plant which were chosen of this study are very important parameters, They play an important part in the planing and management of the national water resources. Most of these records have periods of missing data of the influent BOD5 and TSS. In this study a model for generating missing monthly concentrations influent BOD5 and TSS. Data are introduced. Initially univariate models using the Box-Jenkins approach were fitted to the logarithmically transformed series. Both transformed series were found to be generated by a random process using sampling theory were considered to be white noise. Ordinary regression analysis was performed. No significant correlation between influent BOD5 and TSS concentration were found. KEY-WORDS: Maamera sewage treatment plant, Statistical analysis, stochastic models. فضالت المياه في محطة معالجة مياه المجاري في المعيميرةنماذج عشوائية لبعض خصائص الخالصة رنوعلمم يثرالممفلييثألافاممل يراتغلممخثمثييألاممفيتممفي مماياممحثيثراسمم يلخثامم ييثره ممفضاليرالممفليثرمحمم ميثرلثهامم يراس مم يا فر مم يثرا الممخ لثخ يثراموثخليثرافضلم ي,يسل يأنهفيتا بيلوخثيافافي ماواايثرا ابيثرسلوييرألوكا لنيوثراوثليثر اا يثر فرق ا ظمفيامحلييثرته مل يوثد ثرالفنمفميثرهمهخل يياإلهمف تمفيلخثام ياوللم يي اا امبيثرسلموييرألوكام لنيوثراموثليثر ماا يثر فرقم ر تخثميرالفنفميامقمول يي لهفيثرالفنفم ل ضمفيياوللم يثسمفلييثراتغلمخيافامتهلثفي خلقم ياموكيي لنكلنمزي ثرا امبيثرسلموييرألوكام لنيوثراموثليثر ماا يثر فرقم يرتخكلزثرامقول ي ثرتسممول ميراا امم يثرزانلمم ي ممايكمم يثرتسممول ميراا امم يلو مملياتغلممخيعهمموثضاياعاممتهلثفينامموحدينظممخييل اممخيعنمم ي يوافاممتهلثفي ي ثر اا يثر فرق النيعلفيو وليثختاف يالنيثرتخثكلزيثرلثها يراا ابيثرسلوييرألوكا لنيوثراوثليتيتسال يثالنسلثخ 1. INTRODUCTION Engineers who take up the task of analyzing flows, BOD5, etc. into stream for the purposes of design and planning are often confronted with the problem of working with records having a sequences of missing data. In this study, the readings of influent of BOD5 and influent of Total suspended solid of the wastewater from Maamera sewage treatment plant were considered for the analysis in time series. mailto:nabaalshimmri@yahoo.com Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 187 The infilling of missing values in hydrological data involves the use of statistical procedures of data generation one of such methods is the use of univariate linear stochastic models[Al-Samawi,1986]. Box and Jenkins (univariate models) techniques which were used to determine the appropriate model. These represent the structure of the time series Then tests of these values by the chi-square goodness of fit test and K.S test were performed to check the normality of the data[Hussain,2000]. Many wastewater treatment plants were built within Hilla city. Al-Maamera sewage treatment plant is one of these plants and has begun to operate in 1982. the plant works with an activated sludge system which biologically treats compounds of carbon and nitrogen in raw wastewaters. Maamera sewage treatment plant serve 50000 populations and the treatment facility is a conventional activated sludge system with an average wastewater inflow of 12000 m 3 /day. The sewerage system is designed to accommodate the industrial wastewater, as well as domestic effluent. The treated wastewater in the plant is then being discharged to Shatt Al-Hilla River. A full outline of the plant units is shown in Fig. 1. The objectives of the study is to investigate and analyzed the applicability of such stochastic models to the influent of BOD5 and TSS. in the wastewater of the city of Hilla during the years, 2008 to 2013. 2. METHODS In the present study, certain data have been collected yearly by the Mayoralty of Hilla from the influent in Maamera sewage treatment plant. Major water quality parameters were selected for this study; biochemical oxygen demand (BOD5), Total suspended solids (TSS) over a period of six consecutive years. 3. THEORY The mean of every monthly readings of influent BOD5 and TSS. The parameters that the study depends on the first must be known so that of the time series and its components could be construct. 3.1 Definitions time series model A time series is defined as a set of observations that measure the variation in time of some aspect of a phenomenon, such as the rate of the dissolved oxygen in the stream and the total suspended solids, or the sediment load in a channel [kottegoda, 1980]. 3.2 Components of time series: 3.2.1 Trend: Trend is a steady and regular movement in a time series through which the values are on average either increasing or decreasing. 3.2.2 Periodicity This represents a regular or oscillatory form of variations such as seasonal effect which clearly evident in closely spaced data. In general, the periodic component in a time series can be represented through a system of sin functions after the trend component, if it exists, has been estimated and removed [kottegoda, 1980]. Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 188 3.3 Time series model If a high degree of dependency between sequential observations exists, then forecasting technique which express this dependency and which will generally produce superior results can be applied. These techniques which are presented by Box and Jenkins are called Box – Jenkins model. These techniques are used to identify the appropriate model, other variables and estimate the parameters of the stochastic models. In general; the model are formulated so that the current value of a variable is the weighted sum of past values and a random values which represents the unknown. 3.4 Parameters of the model: 3.4.1 Autocorrelation function(ACF) For series,which are not random, there will be dependency between sequential observations. A useful tool to measure this effect is the autocorrelation function which may be defined as: 𝜌(𝑘) = 𝐸(𝑋𝑡 − 𝑈𝑥 )(𝑋𝑡+𝑘 − 𝑈𝑥 ) √𝐸(𝑋𝑡 − 𝑈𝑥 ) 2. 𝐸(𝑋𝑡+𝑘 − 𝑈𝑥 ) 2 (1) The autocorrelation function has the following properties: 𝜌(0) = 1 |𝜌(𝑘)| ≤ 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘 ≠ 0 𝐴𝑛𝑑 𝜌(𝑘) = 𝜌(−𝑘) For an observed time series 𝑋𝑡 of length N, the autocorrelation function of lag k can be estimated from 𝑟(𝑘) = ∑ (𝑋𝑡 − �̅�)(𝑋𝑡+𝑘 − �̅�) 𝑁−𝐾 𝑡=1 ∑ (𝑋𝑡 − 𝑋) 2𝑁 𝑡=1 (2) Where: �̅� = 1 𝑁 ∑ 𝑋𝑡 (3) 𝑁 𝑡=1 3.4.2 Partial autocorrelation function(PACF) The partial autocorrelation function at lag k is the correlation between Xt and Xt+k with the effects of the intervening observations(Xt+1 ,Xt+2 ,…….,Xt+k-1) removed. [Montgomery and Johnson, 1976]. Notationally, we shall refer to the K th partial autocorrelation coefficient as ∅𝑘,𝑘 The set of parameter ∅1,1, ∅2,2, ∅3,3, which are the last coefficients of the autoregressive models of order 1,2,3,… respectively represent the partial autocorrelation coefficient. A plot of ∅𝑘,𝑘 versus the lag K is called the sample partial autocorrelation function. In general, the partial autocorrelation ∅𝑃,𝑃 is the autocorrelation remaining in the series after fitting a model of order (P-1) and removing the liner dependence. The partial autocorrelation function(PACF) is an important tool in determining the order of the model if the serial correlation function suggests that the process could be approximated by a linear autoregressive model. As a general rule,we would assume a partial autocorrelation coefficient to be zero if the absolute value of its estimate is less than twice its standard error [Kottegoda,1980]. Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 189 3.4.3 Autoregressive processes(AR) The autoregressive processes means that the current observations Xt is "regressed" on previous realizations Xt-1,Xt-2,…Xt-p of the same time series[Montgomery and Johnson,1976]. The autoregressive model AR(P) takes the form. 𝑋𝑡 = ∅𝑃,1𝑋𝑡−1 + ∅𝑃,2𝑋𝑡−2 + ⋯ ∅𝑃,𝑃 𝑋𝑡−𝑃 + 𝑎𝑡 = ∑ ∅𝑃,1𝑋𝑡−1 + ∅𝑡 𝑃 𝑖=1 (4) Where ∅𝑃,𝑖 𝑖 = 1,2,3, … , 𝑃 are the autoregressive parameters or weights and (at) is a white noise process or residuals, the model in eauation(4) is called an autoregressive process of order P, abbreviated AR(P). Also, the model called a linear autoregressive model, in which the current value of a variable is equated to the weighted sum of a (P) number of past values. A variant (at) that is completely random, the word linear merely signifies that the current value is dependent additively upon the past values and not for example, on their squares or square roots [Kottegode, 1980]. 4. RESULTS AND DISCUSSION 4.1 The data The data used in this study are the average of each months for the six-year (2008 – 2013) period for both of influent of BOD5 and TSS into Maamera Sewage Treatment Plant. 4.2 Time plot 2.1 Both of the two Figs (2&3) show that the behavior of the original time series for both influents BOD5 and TSS these Fig show: a. The maximum value for influent BOD5 was (214mg/l) in April 2012. While the minimum value for influent BOD5 was (75mg/l) in Novmaber 2013. b. The maximum value for influent TSS was (301mg/l) in March 2009. And the minimum value for influent TSS was (93mg/l) in March 2010. 2.2 From Figs (4&5) it was noted that, the standard deviation for every year was directly perpotional to the mean in that year. It is noted in the beginning, the standard deviation was low and so was the mean while during the last year the standard deviation became higher with the mean. All these indicated that a logarithmic transformation of the data was needed to stabilize the variance and to make multiplicative effects additive. 4.3 Transformation After adjusting the outlier observation the logarithms for the original time – serieses were taken and are plotted as shown in the Figs (6&7) for both influents BOD5 and TSS these Fig show: a. The standard deviation become constant with the increase of the mean. b. The variation patterns during every year for these series are similar to the variation patterns of the original series. The values for both influent BOD5 and influent TSS are shown in Table1. 4.4 Autocorrelation From Figs (8&9) for influent BOD5 and influent TSS respectively, the autocorrelation function of the series have no trend and seasonality. since the autocorrelation function have the ability of all Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 190 lags are not significant and also,the function have no seasonal cycles[Hipel et al.,1977b], hence the time series has no deterministic for stochastic component From Figs (10&11) for influent BOD5 and influent TSS. It can be that, show the partial autocorrelation functions for two series with confidence limits of (95%). from these four Figs, it can be seen that all autocorrelation coefficient will be within the confidence limits (95%). Hence , it can be said that the two series were (serially independent). 4.5 Test of Normality The test is carried out by two ways: 4.5.1 Chi-Square Test The Chi –Squared statistic depends on specifying the number of histogram classes into which the data will be grouped, and there is no rule that gives the correct number to use [Vose, 2010]. The Chi –Squared test statistic is computed from the relationship: x2 = ∑ (Oi−Ei) 2k i=1 Ei (5) Where 𝑂𝑖 is the observed and 𝐸𝑖 is the expected number of observation in the ith class interval(based on the probability distribution being tested). The expected numbers are calculated by multiplying the expected relative frequency by the total number of observation[Barkotulla et al.,2009].The chi square test parameters are shown in Tables (2) and(3) for influent BOD5 and TSS respectively. From Table (2) it is seen that, the values of 𝑥2 = 7.8508 for influent BOD5 and all the expected frequencies were be larger than or equal to 5[Crof,1979]. The chi-square value is found to be (0.25). This value is within the acceptable region for the normally distributed and that it is white noise series as shown in Fig(12). For influent of TSS, the values of 𝑥2 = 3.5747 and all expected frequencies were greater than(5) as shown in Table (3). The chi-square value was (0.75). This value is within the acceptable region for the normally distributed and that it is white noise series as shown in Fig(13). 4.5.2 Kolmogorov-Smirnov Test Kolmogorov-Smirnov(K-S)goodness of fit test is based on a statistic that measures the deviation of the observed cumulative histogram from the hypothesized cumulative distribution function [Soong, 2004].By using this test, the significant level for influent BOD5 was (0.441), and for influent TSS.was (0.642) as show in Table (4). From all this it can be concluded that the series are white noise and have normal distribution as was obtained from(Chi-square test). 4.6 Regression Analysis The study of regression had done on the three relationships the first relation was between influent of BOD5 and TSS. The data of this relation can be seen from Table (5) and the plot of this relation is shown in Fig(14). Second trial was carried out between the influent BOD5 and the transformed values of TSS, as it seen in Table(6) and Fig(15). Third trial had performed out between the transformed function of influent BOD5 and the transformed values of TSS, as it seen in Table(7) and Fig(16). Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 191 From these three relations, it can be seen that there is no physical relation exist between influent BOD5 and TSS. The values of R 2 for this test were (0.019),( 0.028), and (0.017),respectively. These values were too low to say that the model was adequate for prediction. 4.7 One-Step-ahead-Forecast The forecasting of the sample for influent BOD5 and TSS, during the period of recording the data (2008 to 2013) is depend upon the sampling theory. From the theory of sampling is the estimate of both BOD5 and TSS. Can be found by the following expression: �̅� ± 2𝑆𝑒 (6) Where: �̅�: is the mean for the influent BOD5 and TSS. 𝑆𝑒: is the standard error for the mean influent of BOD5 and TSS. Then for influent BOD5 the forecasting value is (168.19 , 241.71 ) mg/l while for the influent TSS. It is (247.78 , 353.32)mg/l. 5. CONCLUSIONS The following conclusions are drawn from this study: 1. The need for the logarithmic transformation of both influent BOD5 and TSS concentrations data indicates that the two parameters which generate data are non linear in nature. 2. The deterministic component of data of both influent BOD5 and TSS. 3. The time series of both transformed influent BOD5 and TSS is white noise series without residual series. 4. The seasonal effect is not present, so if the time series tales values more than 72 value may be the seasonal effect appear. 5. Box-Jenkins models are not applicable here because the randomness of the data. 6. The forecasting values are derive from the sampling method are tabulated these forecasting values( no each case an interval estimate is given) should be up dated to monitor the values of �̅�and 𝑆𝑒 for each variable(BOD5 and TSS). 7. Relationship between influent BOD5 and TSS concentration: An attempt was made to relate the influent TSS concentration, which is usually easy to measure, with the influent BOD5 which is takes lengther time to determine. The range of possible mathematical relationships covered in this analysis are as follow: (i)The simple linear form, 𝐵𝑂𝐷5 = 𝑎 + 𝑏 𝑇𝑆𝑆 (7) (ii)The inverse form, 𝐵𝑂𝐷5 = �́� + �́�𝑙𝑛 𝑇𝑆𝑆 (8) (iii)The semi inverse form, 𝑙𝑛𝐵𝑂𝐷5 = �́́� + 𝑏 ́́ 𝑙𝑛 𝑇𝑆𝑆 (9) Figs (14),(15) and (16) show the following Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 192 No visual relationship between influent BOD5 and TSS, exists according to the mathematical formulations as given in equations(7),(8) and(9). This finding is supported by the results of the statistical regressions which are tabulated in Table(5),(6) and (7). In all mathematical formulations, the slope coefficients b, �́�, 𝑎𝑛𝑑 𝑏 ́́ were found to be insignificant, thus supporting the findings that no physical relations between influent BOD5 and influent TSS. Hence, the best model which represent the variability of the influent BOD5 is given by the log- normal distribution. Similarly, influent TSS. A concentration may be modeled in the same manner. REFERENCES [1] Al-Samawi, A. A. (1986). "The generation of missing river flow data", Arab Gul,J.of scientific Research. Vol.4, No.2. PP.761-774. [2] Barkotulla M.A.B., Rahman M.S., and Rahman M.M.,(2009). "Characterization and frequency analysis of consecutive days maximum rainfall at Boalia, Rajshahi and Bangladesh", India, Journal of Development and Agricultural Economics Vol. 1(5). PP.121-126. [3] Chatfield, D. (1979). "Statistics for Technolgy", Champaman and Hall,2 nd edition. [4] Croft. D. (1979)."Applied statistics for management studies", Macdonald and Evans, 2 nd edition. [5] Hipel, K, W., Mcleod, A. I. and Lennox, W. C. (1977). "Advances in Box-Jenkins modeling 2. Applications", Water Resources Research, Vol.13,No.3.PP.577-586. [6] Hussain, S. M., (2000). "Time series analysis for hydrochemical pollutants: Applications of Box-Jenkins models to Euphrates river at kufa". M.Sc. Thesis, University of Babylon. [7] Kottegode, N. T., (1980). "Stochastic Water Resources Technology", the macmillan press Ltd. [8] Maidment, D. R. and Parzen, E. (1984). "Cascade model of monthly municipal water use", Water Resources Research, Vol. 20. No. 1, PP. 15-23. [9] Soong T.T., (2004). "Fundamentals of probability and statistics for engineers", John Wiley & Sons Ltd, State University of New York at Buffalo, New York, USA. [10]Vose, David.,(2010). "Fitting distributions to data and why you are probably doing it wrong", 15 Feb 2010, www.vosesoftware.com. http://www.vosesoftware.com/ Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 193 Table (1): Descriptives Parameter Mean Standard Deviation Standard Error BOD5 4.9703 0.2552 0.03095 TSS 5.2185 0.2194 0.02661 Tables (2): Chi-Square Test for the influent BOD5 Lower limit Upper limit Observed frequency Expected frequency Chi-square At or below 4.55 6 5 0.1403 4.55 4.67 6 7 0.1428 4.67 4.79 4 8 2.0075 4.79 4.91 8 10 0.4036 4.91 5.03 13 10 0.8410 5.03 5.15 10 10 0.0022 5.15 5.21 9 6 1.3787 5.21 5.27 4 7 1.2857 Above 5.27 8 5 1.6490 Chi-square=7.8508 with 6 dif, Sig.level=0.25 Tables (3): Chi-Square Test for the influent TSS Lower limit Upper limit Observed frequency Expected frequency Chi-square At or below 4.92 4 5 0.2568 4.92 5.05 9 7 0.4787 5.05 5.13 8 6 0.5565 5.13 5.21 9 13 1.3097 5.21 5.29 14 13 0.0769 5.29 5.37 10 8 0.4199 5.37 5.45 5 6 0.2150 5.45 5.58 5 5 0.0044 Above 5.58 4 5 0.2568 Chi-square= 3.5747 with 6 dif, Sig.level=0.75 Table (4): The values of Kolmogorov-Smirnov Test for all the influents and with confidence level equal 95% Parameters Estimated KOLMOGOROV Statistics DPLVS Estimated KOLMOGOROV Statistics DPLVS Estimated Overall statistics DN Approximate significance level BOD5 0.105 0.085 0.105 0.441 TSS 0.090 0.087 0.090 0.642 Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 194 Table (5): Regression Analysis-Linear Model y=a+bx Dependent variable:BOD5 Independent variable:TSS Parameter Estimate Standard error T-value Probability level Intercept Slope 125.613 0.121 20.906 0.108 6.008 1.119 0.000 0.267 Analysis & Variance Sourc e Sum of square D.f. Mean square F-ratio Probability Mode l Error 1567.889 82599.097 1 66 1567.889 1251.501 1.253 0.267 Total(correlation)= 84166.985 D.f.= 67 Correlation coeffication=0.136 Standard Error Estimate=35.377 R-squared=0.019 Table (6): Regression Analysis-Linear Model y=a`+b`lnx Dependent variable:BOD5 Independent variable: lnTSS Parameter Estimate Standard error T-value Probability level Intercept Slope 8.541 26.822 102.398 19.605 0.083 1.368 0.934 0.176 Analysis & Variance Source Sum of square D.f. Mean square F-ratio Probability Model Error 2321.257 81845.728 1 66 2321.257 1240.087 1.872 0.176 Total(correlation)= 84166.985 D.f.= 67 Correlation coeffication=0.166 Standard Error Estimate=35.215 R-squared=0.028 Table (7): Regression Analysis-Linear Model lny=a``+b``lnx Dependent variable:lnBOD5 Independent variable: lnTSS Parameter Estimate Standard error T-value Probability level Intercept Slope 4.169 0.154 0.741 0.142 5.625 1.082 0.000 0.283 Analysis & Variance Source Sum of square D.f. Mean square F-ratio Probability Model Error 0.076 4.288 1 66 0.076 0.065 1.171 0.283 Total(correlation)= 4.364 D.f.= 67 Correlation coeffication=0.132 Standard Error Estimate=0.25488 R-squared=0.017 Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 195 Figure (1): Image map of Maamera sewage treatment plant, Hilla (Al-Maamera project office, 2012). Figure (2): Time series of influent BOD5 Figure (3): Time series of influent TSS of Al-Maamera S.T.P. of Al-Maamera S.T.P. 60 80 100 120 140 160 180 200 220 240 0 10 20 30 40 50 60 70 80 In fl u e n t o f B O D 5 ( m g /l ) Time(Unit time=monthly period). 80 110 140 170 200 230 260 290 320 0 10 20 30 40 50 60 70 80 In fl u e n t o f T S S ( m g /l ) Time(Unit time=monthly period). (1) Sediment Tanks (2) Reaeration Tanks (3) Primary Stage (4) Office Storages (5) Dilution Tanks (6) New Dilution Tanks (7) New Reaeration Tanks (8) Final Stage (9) Storage Tanks (10) Pump Sediment Stage (11) New Sediment Tanks (12) Compact Unit Station 12 2 4 3 7 11 6 5 8 9 10 1 Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 196 Figure (4): Standard devation of influent Figure (5): Standard devation of influent of BOD5 Vs. Mean. of TSS Vs. Mean. Figure (6): Transformed of Infl. BOD5 series Figure (7): Transformed of Infl. TSS series Figure (8): Autocorrelations for influent Figure (9): Autocorrelations for influent BOD5 series TSS series R² = 0.3375 y = 0.3562x + 1.649 0 20 40 60 80 100 120 0 50 100 150 200 250 S ta n d a rd d e v a ti o n ( m g /l ) Mean (mg/l) y = 0.7878x - 50.66 R² = 0.477 0 50 100 150 200 250 0 50 100 150 200 250 300 350 S ta n d a rd d e v a ti o n ( m g /l ) Mean (mg/l) 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 0 10 20 30 40 50 60 70 80 In fl u e n t o f B O D 5 ( m g /l ) Time(Unit time=monthly period). 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 0 10 20 30 40 50 60 70 80 In fl u e n t o f T S S ( m g /l ) Time (Unit time=monthly period). Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 197 Figure (10): Partial autocorrelations for Figure (11): Partial autocorrelations for influent BOD5 series influent TSS series Figure (12): Frequency Histogram for Figure (13): Frequency Histogram for transformed series of influent BOD5 transformed series of influent TSS Al-Qadisiyah Journal For Engineering Sciences, Vol. 8……No. 2 ….2015 198 Figure (14): Regression of influent of BOD5 on the influent of TSS. Figure (15): Regression of influent of BOD5 on the logarithm transformed influent of TSS. Figure (16): Regression of Transformed logarithm of influent of BOD5 on the logarithm transformed influent of TSS. 50 75 100 125 150 175 200 225 250 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 In fl u e n t o f B O D 5 ( m g /l ) Transformed logarithm of TSS (mg/l) 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 T ra n sf o rm e d l o g a ri th m o f in fl u e n t o f B O D 5 Transformed logarithm of influent of TSS. 50 75 100 125 150 175 200 225 250 50 75 100 125 150 175 200 225 250 275 300 325 In fl u e n t o f B O D 5 ( m g /l ) Influent of TSS (mg/l).