#### Abstract

This study is part of the development of a model of hydraulic calculation of controlling reserves control in drinking water supply systems. To that end, the general expressions of the cumulative curves of gravity water supply and pumped water supply as well as the ones of the distribution over time were formulated. It appears from this study that for a peak coefficient Kp between 1.2 and 1.4, the cumulative consumption evolves linearly and for Kp ranging from 1.45 to 2.5 the cumulative consumption follows a polynomial curve of degree 3. Then, there is a relationship between the respective totals of gravity water supply and pumped water supply and the total of the cumulative consumption, which superimposed have enabled the establish- ment of the mathematical model for determining the volume of the controlling reserves.

INTRODUCTION In drinking water supply systems, drinking water reservoirs are needed in cities where sponta- neous urban settlements with middle class standing are important. The satisfaction of the pop- ulation with respect to the water demand is ensured by the continuous service offered by the structures and equipment of the water supply system throughout the day. This performance is largely ensured by water supply reserves (such as cistern, water towers etc.) that serve as a buffer between the production which is fairly constant and the distribution which remains highly variable with time. Poor design of these reservoirs in water networks causes severe problems on the performance and even on the cost of the entire water supply network [4]. The various reserve components (regulation, fire, emergency and maintenance) of the reservoir volume show that the storage capacity varies depending on the controlling reserves since the others are estimated with specific standards or guidelines for each region. Controlling reserves for a given day represent about 20% of the daily consumption, the maxi- mum volume of control being calculated from the hourly consumption of the day of the high- est consumption [5]. However, studies have shown that according to guidelines,the reservoir capacity can range from less than 25% to 100% of the daily consumption peak taken for the project horizon [2.9]. Indeed, the present study focuses on the development of a rational ap- proach for the hydraulic calculation of the controlling reserves to reduce differences on the basis of the analysis of design approaches currently used by field engineers. 2. METHODOLOGY 1.1 Materials The cumulative water supply has been modelled using the Laplace transform which is a widely used tool for mathematical modelling of physical phenomena (inflow, consumption pattern, etc.). The Maple 13 software enabled the adjustments of curves of the cumulativecon- sumption for each peak coefficient. MATLAB software is used for the treatment of matrix components of the pumped water flow rate and the pumping time. 2.2 Methods The most used method to estimate the controlling volume allows viewing offs between theperiodsof low consumption and those of high consumption in order to adjust the pumping periods to minimize the risk of rupture of supply during hours of high consumption [5]. In fact, we fix the time for daily pumping, the pumping periods and the pumping rate and we successively represent per day the water supply and the consumption, simplified in hourly slot, the curves of the accumulated previous flow rates and the superposition of the curves of ac- cumulated flow rates. A parallel translation of the supply curve for covering the distribution curve allows the visualization of both maximum fluctuations. The sum of these two fluctua- tions indicates the volume of the controlling reserves. These methods have been generally replaced by extended period simulation models which can be more flexible than the graphical methods. Reservoirs operation Analysis showed that the capacity of the reservation distribution is function of the supply flow rate and the fluctuations in the distribution flow rate [8]. Let the hourly flow rate of water supply, the hourly flow rate of distribution and theflow rate of controlling reserves, then: 2.2.1 Modelling of the water supply Pumped water supply The function reflects the pumped flow rate at time t, definesas [10]: , where:t is the pumping time, , is the pumped flow rate in the nth time inter- val, is the number of intervals (n 1). To obtain the Laplace transform [7] of the function which allows for the accumulation of supply, it is necessary to express it in the form of linear combination of unit step functions: hence : With - the Heaviside function defines as: Let be the cumulative flow rates of supply, . Using the theorem of the Laplace derivations [7], it is established that: Whereas , since there is no pumping at time From equation (2), we have: . Whereas the Laplace transform of the Heaviside function gives: From equations (4) and (5), we obtain the Laplace transform of : The inverse Laplace transform can enable to express as Whereas = With , where , here the exponent is equal to 1 according to the equation (8) then , and then: From equations (7) and (9), we obtain the accumulation of supply as follows: The expression (10) is the model of accumulation of pumped water supply over time. Gravity water supply The function reflects the flow rate of gravity water supply at time t, definesas: We have: (n being the number of intervals) ; , The expression (11) shows that the accumulation of water supply is linear with respect to time. 2 Modelling consumption Modelling the consumption pattern is based on the hourly distribution of the maximum daily consumption according to the hourly peak coefficient Kp ranging from 1.2 to 2.5 [2] (Fig. 1). The consumption curves (Figure 2) were used to calculate the cumulative consumption based on the hourly peak coefficients Kp. It is observed that the cumulative consumption var- ies linearly for Kp between 1.2 and 1.4, and varies in the form of polynomial curve of degree 3 for Kp ranging from 1.45 to 2.5. Figure 1: Consumption Curves over 24 hours (in% of the maximum daily consumption) Figure 2:Evolution of the cumulative consumption for different hourly peak coefficients ac- cording to the time The expression of the accumulated consumption becomes then: - for 1,2 ? Kp ? 1,4, where - Cumulative consumption at time t, m3; - maximum daily consumption, m3 / day; A and B - linear adjustment coefficients recorded in Table 1, For 1.45 ? Kp ? 2.5, where A ', B', C '- polynomial fit coefficients of degree 3, recorded in Table 2. 3 Determining the controlling reserves Based on equations (10) and (11), it is observed that the cumulative water supply almost follow a straight trend linefor a given time period. The cumulative consumption varies greatly during the day and depends on the variation of the hourly peak coefficient as indicated by equations (12) and (13). The controlling reserveswhich is a buffer between the cumulative supply during the day and the daily distribution, by the accumulation of excess water during periods of low consumption and its restitution during hours of high consumption. Its expression is therefore as follows: ? Expression of X (t) in the case of a gravity water supply From equations (11), (12) and (14), we have: From equations (11), (13) and (14), we have: ? Expression of X (t) in the case of pumped water supply Form equations (10), (12) and (14), we have: Form equations (10), (13) and (14), we have: The controlling reserves are obtained from the superposition of the curves of flow rates of cumulative water supply and the ones of the distribution. A parallel translation of the sup- ply curve for covering the distribution curve allows the visualization of both maximum fluc- tuations. The sum of these two fluctuations indicates the minimum volume of the controlling reserves. 4 RESULTS AND DISCUSSIONS The minimum volume of the controlling reserves, on the basis of equations (14) and (15), is the sum, in absolute values, of the largest value and smallest (negative) value of X (t). 4.1 Gravity water supply From equation (15) and table 1, we have: ? whereas , for all values of A, hence . With - the limit time between filling up and the supplying of water to the network linked to the reservoir. for ,then is greater and it occurs the double supply of water to the network by the reservoir and source. By framing on this interval, we obtain its smallest negative value which is as follows: for , at that time of the day occurs the filling up of the reservoir. Simi- larly, framing on thisinterval enables the finding of its maximum positive value which is as follows: 20) The minimum volume of controllingreservesbecomes: Equations (19) and (20) give: with where - the volume correction factor which corrects the adjustment differences, - the cor- rection coefficient obtained by simulation with a MATLAB programme which adjusts the volume of the controlling reserves. It is based on the hourly peak coefficient in Figure 3. In the conditions related to (16), being a third degree polynomial, its maximum and minimum values are determined from the properties of its first and second derivatives. So: , (24) ? . Based on table 2, so there are two distinct solutions and of : and . Seeking the maximum and minimum of , let us integrate and in the second derivative of , we have: , then has a minimum at point which represents the smallest negative value, then has a maximum at point , which represents the largest positive value. From equation (21), the volume of the controlling reserves becomes: ,then Figure 3.a shows that the coefficient is equal to 0 for Kp = 1.2 and reaches its maximum value equal to 1.8 at Kp = 1.32 and then decreases to 1 for Kp = 1.4. However, the volume of controlling reserves calculated with equation (21) is suitable for water supply conditions in cities with medium standing for equal to 0,018 . Figure 3.b shows that the coefficient increases from 1 for Kp = 1.45 to 2.4 forKp = 1.9 by following a gradually varied progression where it reaches its maximum value at 2.4. It tends to infinity for Kp between 2.3 and 2.4, and vanishes at Kp = 2.16 ( ) then de- creases down to -0.9. However, the sum of the differences expressed above decreases forcities with low population whose urbanization is spontaneous and for whichKp varies from 2.16 to 2.3. In fact, the expression (25) of the volume of controlling reserves is suitable for Kp be- tween 1.45 to 2.3 with a maximum volume correction equal 0,024 . 4.1 Pumped water supply In this case, two aspects are considered: the supplied flow rate is spread over 24 hours of the day and so it is constant and equal to or the supplied flow rate is variable and in fact enables to reduce the required volume of reservoir (which is important especially in the case of an elevated reservoir). The maximum hourly flow rate provided by the pump station thus depends on the selected operating mode; it is generally between (for uniform pumping) and . Therefore, the correction volume is set equal to 0. The development of equation (10) by iterative method provides its simplified form: where - difference between the upper bound and the lower bound of the ith interval. By incorporating equation (26) into equation (10), we proceed to the framing of (t) for t , and we have: Here, the fact that the pumping rate is variable from one interval to another, equation (17) becomes then: From equation (28), we proceed to the framing of (t) for t , and we have: From equations (14) (27) (29) and 1,2? Kp ? 1.4, we proceed to the framing of on . This yields to: So : The values of I and J are respectively the maximaon each time interval considered. Similarly to the considerations leading to the expression (28), equation (18) is for and 1,45 Kp 2,5: , So : , and . By setting: , we have : If and then admits only a positive maximum on and , (37) where - the largest positive difference between the totals of supply and consumption on the nth interval, - the smallest gap on the nth interval. In this case, there is filling of the reservoir during this time slot. If and then X (t) admits a negative maximum on and , (38) If and then has a positive maximum and a negative maximum on , , and . (39) If and then has a positive maximum and a negative maximum on and . (40) The minimum volume of the controlling reserves in the reservoir becomes the sum of the maxima of differences and related to the conditions (37) (40) on the intervals spread over 24 hours of the day, and then the expression of becomes: . (41) This approach shows that thecapacity of the controlling reservesis defined using the inte- gral curves whose ordinates give the quantity of the cumulative water consumed since the beginning of the day until the end of each specified time. Applying this calculation model on drinking water systems in cities with spontaneous ur- banization and medium standing in Benin (Figure 5) shows that the capacity of the controlling reserves in the reservoirs is 2.5 to 6 % of the daily peak consumption for a pumping system in steps and 15 to 30% when the pumping system is uniform throughout the day, which is close to the rate proposed by other researchers [2, 6, 9]. 5 CONCLUSION The general expressions of curves of the cumulative gravity water supply and pumped water supply as well as the distribution over time formulated using the Laplace transform, have enabled the determination of the volume of the controlling reserves from the daily peak consumption, the pumping system and the hourly peak coefficient. The cumulative consump- tion scales linearly for a peak coefficient Kp between 1.2 and 1.4, and follows a polynomial curve of degree 3 for Kp ranging from 1.45 to 2.5. It is established the relationship between the totals respectively gravity supply systems and delivery and the cumulative consumption, which superimposed have enabled the finding of the mathematical model for determining the control volume ranging from 3 to 30% of the daily peak consumption. a) b) Figure 4: Hourly consumption of the peak day for hourly peak coefficients a) and b)

### Taofic BACHAROU

Polytechnic School of Abomey-Calavi, University of Abomey-Calavi, Laboratory of Energetics and Applied Mechanics

Email: btaofic@yahoo.fr

01 BP 2016 Cotonou, Benin

### Gossou HOUINOU

Polytechnic School of Abomey-Calavi, University of Abomey-Calavi, Laboratory of Energetics and Applied Mechanics

01 BP 2016 Cotonou, Benin

### Gerard A GBAGUIDI

Polytechnic School of Abomey-Calavi, University of Abomey-Calavi, Laboratory of Energetics and Applied Mechanics

01 BP 2016 Cotonou, Benin

### Clement AHOUANNOU

Polytechnic School of Abomey-Calavi, University of Abomey-Calavi, Laboratory of Energetics and Applied Mechanics

01 BP 2016 Cotonou, Benin

### Ezechias KANHO

Polytechnic School of Abomey-Calavi, University of Abomey-Calavi, Laboratory of Energetics and Applied Mechanics

01 BP 2016 Cotonou, Benin

### Ruben AZA-GNANDJI

Polytechnic School of Abomey-Calavi, University of Abomey-Calavi, Laboratory of Energetics and Applied Mechanics

01 BP 2016 Cotonou, Benin

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