Optimizing pumping rate in pipe networks supplied by groundwater sources
- 1. KSCE Journal of Civil Engineering (2013) 17(5):1179-1187 DOI 10.1007/s12205-013-0116-4 − 1179 − www.springer.com/12205 Water Engineering Optimizing Pumping Rate in Pipe Networks Supplied by Groundwater Sources Mohamed M. Somaida*, Medhat M. H. El-Zahar**, Yasser A. Hamed***, and Mahmoud S. Sharaan**** Received March 2, 2012/Revised July 21, 2012/Accepted November 12, 2012 ·································································································································································································································· Abstract In the present paper, an analytical solution has been reached for determining the optimum pumping rate in a pipe network supplied from a ground water source by means of water wells. The solution depends on the use of the gradient technique which requires the derivatives of the studied cost functions and is based on the maximum economic benefit from the produced water. The solution is examined on a predetermined optimal water distribution system, and gives values for optimum pumping rates which are in good agreement with those obtained using the graphical solutions, indicating the reliability of the analytical solution. The effect of the initial cost on the optimum pumping rate and its bearing on the unit price of water has been also studied. This reflects the importance of choosing the proper unit water price that makes the water return covers the global cost and satisfies the maximum economic benefit from the produced water. Studying the effect of unit price of water on the water return or different pumping rates shows that, whether ignoring or including the initial cost, the water return increases with increase of pumping rate till reaching a turning point, often which the water return begin to drop slightly. Hence, this point determines the optimum pumping rate in the pipe network as well as the maximum economic benefit satisfied. Finally within the scope of the present study, it is concluded that, the identity or good agreement found in the results of optimum pumping rates, indicates the validity of the developed analytical solution and the employed graphical solutions. Keywords: groundwater, pipe network, pumping rate, water distribution, water income ·································································································································································································································· 1. Introduction In most of the studies concerning water-distribution systems supplied by pumped groundwater sources, little attention is paid to the cost of wells, optimal pumping rates and water return. The desired pumping rate in a pipe network supplied from a groundwater source may be selected on the basis of effective demand, maximum well yield or maximum economic benefit. Selection on the basis of demand is done when the required flow in the network is less than the capacity of wells. Maximum pumping rate is necessary when the demand equals or exceeds the well capacity. Selecting the pumping rate on the basis of maximum economic income is possible when the wells do not have to meet a particular demand and when the difference between the water return and the total cost is greatest. This optimization problem will be discussed here below. 2. Background The majority of studies dealing with groundwater have employed optimization techniques for groundwater withdrawal. Tung (1986), assembled a model that searched for optimum pumping rate in a given well pattern to minimize drawdown. The model combined the Cooper-Jacob equation for well drawdown and a linear programming algorithm. Calborn (1991), assembled a simulation model which provides an estimate of the best combination wells to meet a particular flow rate. Bouwer (1978), determined the optimum well discharge for a hypothetical pumping well directly to a storage tank using a graphical procedure. In his study, he ignored the well construction costs, other fixed cost and the hydrologic parameters of the aquifer. Somaida (1991), studied the same problem for a given well example using analytical and graphical solutions. Also, he studied the effects of hydrologic, hydraulic and cost parameters on the optimum well discharge. Somaida (1993), studied the effects of the hydraulic and cost parameters on the economical optimum well discharge supplied from a groundwater source. Also, the problem of pumping optimization in coastal aquifers are studied, Featherstone et al. (1983), Fetter (1994), Naggar (2002), these studies must be treated with caution, since coastal aquifers are highly sensitive to disturbances and in *Professor, Hydraulics, Civil Engineering Dept., Port Said University, Port Fouad, Port Said 42523, Egypt (E-mail: medosensi@gmail.com) **Assistant Professor, Sanitary Engineering Civil Engineering Dept., Port Said University, Port Fouad, Port Said 42523, Egypt (Corresponding Author, E- mail: melzahar@yahoo.com) ***Associate Professor, Irrigation and Hydraulics Civil Engineering Dept., Port Said University, Port Fouad, Port Said 42523, Egypt (E-mail: yasser_ham@hotmail.com) ****Graduate Student, Civil Engineering Dept., Suez Canal University, Ismailia 41522, Egypt (E-mail: mah_samy2010@yahoo.com)
- 2. Mohamed M. Somaida, Medhat M. H. El-Zahar, Yasser A. Hamed, and Mahmoud S. Sharaan − 1180 − KSCE Journal of Civil Engineering appropriate management may lead to their destruction as a some of fresh water due to the intrusion of salt water from the sea, Naggar (2002). Concerning with unconfined phreatic aquifers, this research deals with the mathematical optimization of pumping rate in pipe networks supplied from phreatic unconfined aquifer by water wells using the gradient technique. The main objective is the determination of the optimum pumping rate through the water distribution system using a developed analytical solution, which satisfies the maximum economic benefit from the produced water. 3. Proposed Study The present study utilizes the same aquifer system and the same suggested pipe network given in, Somaida, 2011. The suggested pipe network layout is shown in Fig. 1. The pipe network consists of two wells, a Pump Well (PW) and a Test Well (TW). Both wells are connected to a junction, which is connected to an Elevated Tank (ET), then a 674.5 m pipeline from the elevated tank to the inlet of the pipe network. 3.1 Groundwater Source In the aquifer example, groundwater occurs under water table conditions. The water-bearing stratum is a thick body of medium and coarse sand mixes, Fig. 2. Most of the recharge to the aquifer is by precipitation and infiltration from streams. The aquifer can provide sufficient quantities of water which can be used for domestic consumption. 3.1.1 Analysis of Step-Drawdown Test The result of drawdown test conducted in the same well (TW) are given in Table 1, Rajput (1998). The data obtained from the step-drawdown test are analyzed as shown in Fig. 3 by plotting. The equation of straight line shown in Fig. 3 is: Sw = Cf Q + Cw Q2 Sw = 0.0004 Q + 3 × 10−7 Q2 (1) Where, Cf = Formation loss coefficient = vertical intercept of straight line (0.0004 day/m2 ), and Cw = well loss coefficient = slope of the straight line (3 × 10−7 day2 /m5 ) Q= Well discharge (m3 /day) Sw = Drawdown in the well (m) The above equation is useful for determining drawdown Sw or any particular discharge from well (Q). 3.1.2 Optimized Pipe Network under Study Two separate optimized pipe networks, one of equal diameters and other one of unequal diameters are taken from Somaida (1991). Their optimized commercial diameters were determined from the initial diameters using the derivative method, Somaida (1991), Somaida (1993), Somaida (2011). The results of the optimized commercial diameters are given in Table 2. Where: N35 PNW is the lowest cost equal diameter pipe network, and Sch (3) PNW is the lowest cost unequal diameter pipe network. It is obvious that Sch (3) PNW is more profitable than N35 PNW 3.2 Analytical Solution for Pumping Rate The formulae for total cost and cost parameters are given here below, all after Somaida (2011): Total Cost = CT = Cpi2 + Cp1 + Cp2 + Cwe + Cet + CLm +Ctr + Cin (2) Fig. 1. Suggested Layout of Pipe Network and Setting of Pump, Elevated Tank, Main Pipes and PNW Fig. 2. Section of the Aquifer Studied Table 1. Results of Step Drawdown Test Discharge, Qw (m3 /day ) 1382.4 2131.2 3110.4 4636.8 6264 Drawdown, Sw (m) 1.32 2.13 3.66 9.73 13.8 Fig. 3. Specific Drawdown versus Discharge of the Test Well
- 3. Optimizing Pumping Rate in Pipe Networks Supplied by Groundwater Sources Vol. 17, No. 5 / July 2013 − 1181 − Where, (1) Cpi2 = Pipe network cost = a × L × D2.12 (LE) L = length of pipe (m), D = diameter of pipe (m), and a = pipe cost coefficient taken according to the Egyptian local market for PVC pipes = 1423 or Cpi2 = 1423 × L × D2.12 (LE) (2) Cp1 = Pumping cost = A1 × Qt × Hd Qt = pumping rate (m3 /day) from both wells, Hd = operating head of pump (m) and A1 = unit cost of pumping = 2.984 LE/ (m3 /day)/mLift ($0.5), which is based on an average daily pumping of 12 hours, life period of scheme of 1.5 years, 0.28 LE/KW hour and pump efficiency of 0.7, (9). (3) Cp2 = pump cost = A2 × Qt × Hd (LE) A2 = unit cost of pump = 0.6 LE/(m3 /day)/ mLift ($0.1), based on a cost of pump/KW of 3900 LE and a pump efficiency of 0.7, (9). (4) Cwe = well cost = 160,000 LE ($26650) for two wells (5) Cet = elevated tank cost (LE) = C × Qt 0.77 C = cost coefficient for concrete elevated tank = 450 according to the Egyptian local market (6) CLm = labor and maintenance cost (LE) = C1 × Qt 0.47 Qt in m3 /day and C1 = cost coefficient of labor and maintenance (5000), based on average daily pumping of 12 hours and 15 years life period of scheme. (7) Ctr = treatment cost (LE) = C2 × Qt 0.65 Qt in m3 /day, C2 = cost coefficient of treatment (520), based on the same assumptions in (6). (8) Cin = inclusive cost (LE) = C3 × Qt 0.54 Qt in m3 /day, C3 = inclusive cost coefficient (3100), based on the same previous assumptions as in (6) and (7). This analysis includes the cost of the main pipe network only. The initial cost of the well and the elevated tank is excluded with the previous equations repeated as a function in the pumping rate Qt. 3.2.1 For Pipe Network Cpi2 = 1423 × L × D2.12 where, For all pipes, (3) Where, D= Diameter of pipe (m) fi = Pipe friction factor (function of K/D and Reynolds number) K= Roughness height, and Li = iNdividual pipe length (m) m= Number of pipes in pipe network (12) Qi = Individual pipe discharge (m3 /day) Qt = Pumping rate through PNW (m3 /day) S= Hydraulic gradient 3.2.2 For Pumping and Pump Cp1 + Cp2 = (A1+A2) · Qt · Hd Where A1, A2 and Qt are previously defined. Hd = Operating head of pump which according to Fig. 1 is given by: Hd = Sw + Z + hr + hL1 + hL2 + hfmax Where, CPi2 1423 Li 0.33 fi Qi 2 × × S 24 60 × 60 × ( ) 2 × ----------------------------------------- - ⎝ ⎠ ⎛ ⎞ 0.424 × × = D 0.33 fi × Qi 2 × S 24 60 × 60 × ( ) 2 × ------------------------------------------ ⎝ ⎠ ⎛ ⎞ 0.2 = CPi2 0.0579 Li f i 0.424 × Qi 0.848 × S 0.424 ------------------------------------- - ⎝ ⎠ ⎛ ⎞ i 1 = i m = ∑ × = CPi2 0.0579 Li f i 0.424 × mi 0.848 × Qt 0.848 × S 0.424 ------------------------------------------------------- - ⎝ ⎠ ⎛ ⎞ i 1 = i m = ∑ × = Table 2. Diameters and Costs for the Initial and Optimal Networks according to the Derivative Method Pipe Ref. No. Length (m) Initial Diameter (mm) Derivative Method Initial Diameter (mm) Derivative Method N35 P.N.W Commercial diameters, (mm) Sch. (3) P.N.W Commercial diameters, (mm) P2 179 350 296.6 350 296.6 P3 547 350 296.6 350 296.6 P4 281 350 296.6 350 296.6 P5 236.5 350 211.8 350 235.4 P6 254 350 188.2 300 211.8 P7 362.5 350 188.2 300 211.8 P8 363.5 350 188.2 300 188.2 P9 338 350 188.2 300 188.2 P12 257.5 350 235.4 300 211.8 P13 345.5 350 188.2 200 150.6 P14 233.5 350 150.6 200 103.6 P15 235 350 103.6 150 103.6 Total Cost (LE) 3,917,892.0 3,521,212.0 3,770,583.3 3,520,818.5 Total Cost ($) 652,982.0 586,868.67 628,430.5 586,803.08
- 4. Mohamed M. Somaida, Medhat M. H. El-Zahar, Yasser A. Hamed, and Mahmoud S. Sharaan − 1182 − KSCE Journal of Civil Engineering Cf = Formation loss coefficient (day/m2 ) Cw = Well loss coefficient (day2 /m5 ) hL1 = Friction and minor losses in the main from the well to ET = k1 · rQt 2 (m) hL2 = Friction and minor losses in the main from ET to the inlet of pipe network = k2 · Qt 2 (m) and, hfmax = Maximum head lost by friction encountered in the pipe network (m) hr = Residual pressure head in the pipe network (m) Qw = Discharge per well (m3 /day) Sw = Well drawdown (m) Z = Depth of water table from ground level before pumping (m) Thus, (4) Substituting the cost parameters given above in Eqs. (3) and (4), in the total cost equation and rearranging, then: (5) The optimum pumping rate is that at which the difference between Water Return (WR) and the total cost CT is greatest, i.e.,: WR- CT = maximum (6) The water return is calculated by the following equation. WR = b (365 × Y × r × Q) = B × Qt Where: b= Unit price of water produced, (LE/m3 ) ($/m3 ) B= Unit water return, (LE/m3 /day) ($/m3 /day) B= b (365 × Y × r) = 0.100 × 365 × 15 × 0.5 = 273.75 LE/m3 /day = $45.398 /m3 /day Qt = daily pumping rate, (m3 /day) r= Percent daily of pumping (12/24 = 0.5) Y= Life period of Scheme (15 year) Then Eq. (6) will take the form: (7) Differentiating Eq. (7), with respect to Qt and eliminating the fixed cost parameter (initial cost) and equating with zero, then: (8) The previous equation can be written as follows: (9) Values of coefficients in Eq. (9) are listed as follows: A1 = Unit cost of pumping, A1 = 2.984 LE / (m3 /day) / m lift = $0.495 / (m3 /day) / m lift A2 = Unit cost of pump, A2 0.6 LE / (m3 /day) / m lift = $0.095 / (m3 /day) / m lift t Cf = The formation loss coefficient (0.0004 day/m2 ) - Cw = the well loss coefficient (3 × 10−7 day2 /m5 ) C2 = The coefficient of treatment cost, (520) – and, C3 = the coefficient of inclusive cost, (3100) Hd = Operating head of the pump (m) - C1 = the coefficient of labor and maintenance costs, (5000) (K1+K2)= Friction and minor loss coefficients in the main Substituting the numerical values of the previous coefficients in Eq. (9) and rearranging the equation, then: (10) Eq. (10) is based on the following assumptions: (1) The use of optimized networks according to (9). (2) Using the gradient technique which requires the derivative of the studied cost functions. (3) The cost of pipe networks, pump, pumping, friction losses, well cost, operating costs, and water return are considered. (4) The pipe costs are evaluated as function of pumping rate. (5) The initial costs of water wells and elevated tan are consid- ered fixed, in addition to the cost of main. (6) The wells supplied from the groundwater source (a phreatic unconfined aquifer), don't have to meet a specified demand. (7) The unit price of water must be selected such that water CP1 CP2 + A1 A2 + ( ) ⋅ = Cf 2 ---- - Qt 2 ⋅ Cw 4 ----- - Qt 3 ⋅ Z Qt ⋅ hr Qt K1 K2 + ( ) Qt 3 hf Qt ⋅ + ⋅ + ⋅ + + + ⎝ ⎠ ⎛ ⎞ CT 0.0579 Li fi 0.424 mi 0.848 Qt 0.848 ⋅ ⋅ ⋅ S 0.424 ------------------------------------------------- - ⎝ ⎠ ⎛ ⎞ i 1 = i m = ∑ × = A1 A2 + ( ) cf 2 ---Qt 2 cw 4 ---- -Qt 3 ZQt hrQt K1 K2 + ( )Qt 3 hfQt + + + + + ⎝ ⎠ ⎛ ⎞ + C1Qt 0.47 C2Qt 0.65 C3Qt 0.54 + + + B Qi 0.0579 Li fi 0.424 mi 0.848 Qt 0.848 ⋅ ⋅ ⋅ S 0.424 ------------------------------------------------- - ⎝ ⎠ ⎛ ⎞ i 1 = i m = ∑ × – × A1 A2 + ( ) cf 2 ---Qt 2 cw 4 ---- -Qt 3 ZQt hrQt K1 K2 + ( )Qt 3 hfQt + + + + + ⎝ ⎠ ⎛ ⎞ + C1Qt 0.47 C2Qt 0.65 C3Qt 0.54 + + + max = B A1 A2 + ( ) -------------------- 0.0491 A1 A2 + ( ) -------------------- Li fi 0.424 mi 0.848 Qt 0.152 – ⋅ ⋅ ⋅ S 0.424 --------------------------------------------------- ⎝ ⎠ ⎛ ⎞ i 1 = i m = ∑ × = cf Qt 3cw 4 ------- -Qt 2 Z hr 3 K1 K2 + ( )Qt 2 hf + + + + + + ⎝ ⎠ ⎛ ⎞ + 1 A1 A2 + ( ) -------------------- 0.47 C1Qt 0.53 – × 0.65 C2Qt 0.35 – 0.54 C3Qt 0.46 – × + × + ( ) + B A1 A2 + ( ) -------------------- 0.0491 A1 A2 + ( ) -------------------- Lf f i 0.424 mi 0.848 Qt 0.152 – ⋅ ⋅ ⋅ S 0.424 --------------------------------------------------- - ⎝ ⎠ ⎛ ⎞ i 1 = i m = ∑ × = Cf 2 ---- -Qt Cw 2 ----- -Qt 2 Hd 2 K1 K2 + ( )Qt 2 + ( ) + + + 1 A1 A2 + ( ) -------------------- 0.47 C1 Qt 0.53 – 0.65 C2 Qt 0.35 – 0.54 C3 Qt 0.46 – ⋅ × + ⋅ × + ⋅ × ( ) + ≈ B A1 A2 + ( ) -------------------- 0.0137 Lt f t 0.424 mi 0.848 Qt 0.152 – ⋅ ⋅ ⋅ S 0.424 --------------------------------------------------- - ⎝ ⎠ ⎛ ⎞ i 1 = i m = ∑ × = Cf 2 ---- -Qt Cw 2 ----- -Qt 2 Hd 2 K1 2K + ( ) Qt 2 ⋅ + + + + 655.69 Qt 0.53 – × 94.308 Qt 0.35 – × 467.07Qt 0.46 – + + ( ) +
- 5. Optimizing Pumping Rate in Pipe Networks Supplied by Groundwater Sources Vol. 17, No. 5 / July 2013 − 1183 − return covers the global cost. (8) The utilized optimum pumping rate feeding the pipe net- work by water wells should not exceed the safe yield of the ground water source for fear of deterioration of the aquifer. (9) Steady flow conditions, i.e., discharge from the ground water source are compensated by an equal recharge from streams. 3.3 Application 3.3.1 Optimum Pumping Rate Results The results of optimum pumping rate obtained from the solution of the developed equation for each case are shown in Table 3, at unit price of water b = 0.1 LE/m3 , ($1.0 = 6.0LE) As shown in Table 3, the optimum pumping rate obtained is equal to 6,989.1 and 6,880.1 m3 /day for each of optimized commercial N35 and Sch (3) PNW, respectively. 3.3.2 Effectiveness of the Results To check the validity of the developed equation for optimum pumping rate, a graphical solution can be used as follows. First, the WR line is drawn at different pumping rates for each of the pipe networks studied. The income line is then drawn with the same scale at b = 0.1 LE/m3 . The optimum discharge is obtained when the vertical distance between the cost line and WR line is greatest, indicating a maximum economic income, Figs. 4, 5. The figures show that at a unit price of water of 0.1 LE/m3 , the optimum discharge is 6,200 m3 /day for optimized commercial N35 PNW and 6,000 m3 /day for optimized commercial Sch. (3) PNW, Table 4. Correlation of the optimum discharges given in Tables 3 and 4 shows that the results of optimum discharge using the developed equation are of nearer magnitudes and in good agreement with those obtained by the graphical solutions. This indicates the effectiveness of the developed equation, the given cost parameters, and the assumptions. Also, the b values are reasonable and may agree with the current pricing of water. 3.4 Effect of Initial Cost on Optimum Pumping Rate 3.4.1 Excluding Initial Cost The graphical solution is used to study the effect of excluding the initial cost of wells and elevated tank on the optimum pumping rate at different unit price of water. This is applied on Sch. (3) PNW which is of optimized commercial diameters and has the least minimum cost, Table 2, the unit price of water is set at the current Egyptian water price, which ranges from 0.1 to 0.2 LE/m3 . The results of optimum pumping rate at different unit prices of water are shown in Fig. 6. 3.4.2 Comparison between the Analytical and Graphical Solutions for Optimum Pumping Rate The results for optimum Qt obtained by the graphical and analytical solutions at different unit price of water b are shown in Table 5 for optimized commercial Sch. (3). Correlation of Fig. 6 and Table 5 shows that as the unit price of Table 3. Optimum Pumping Rate, Unit Water Return and Unit Price of Water Calculated according to the Mathematical Solution Optimized Commercial Diameters Optimum pumping rate Qt, (m3 /day) Unit water return B, (LE/(m3 /day)) Unit price of water b, (LE/m3 ) N35 P.N.W 6,989.1 273.75 0.100 Sch.(3) P.N.W 6,880.1 273.75 0.100 Fig. 4. Optimum Pumping Rate Qt, Optimum Commercial N35 PNW, at b=0.1 LE/m3 ($1.0=6.0 LE) Fig. 5. Optimum Pumping Rate Qt, Optimum Commercial Sch. (3) PNW, at b=0.1 LE/m3 ($1.0=6.0 LE) Table 4. Optimum Pumping Rate, Unit Water Return and Unit Price of Water Calculated according to Graphical Solution Network Optimum Pumping Rate Qt, (m3 /day) Unit Water Return B, (LE/(m3 /day)) Unit Price of Water b, (LE/m3 ) N35 P.N.W 6,200.0 273.75 0.100 Sch. (3) P.N.W 6,000.0 273.75 0.100 Fig. 6. Optimum Pumping Rate at Different Unit Price of Water Excluding Initial Cost (opt. comm. Sch. (3))
- 6. Mohamed M. Somaida, Medhat M. H. El-Zahar, Yasser A. Hamed, and Mahmoud S. Sharaan − 1184 − KSCE Journal of Civil Engineering water increases, the optimum Qt increases. In addition, the results for optimum Qt, using the analytical solution are of nearer magnitudes with those obtained by the graphical solution, indicating the validity of the analytical solution. 3.4.3 Including the Well Cost A graphical solution given by Somaida (1993), is employed to study the effect of including the well costs on the optimum pumping rate at different unit price of water. It is summarized as follows: (1) First, determine the point of fixed well costs at Qt = zero which is taken as the origin, Fig. 7. (2) Starting from the origin, draw the cost line and water return lines given in Fig. 6, (excluding well costs), or the same scale and trends as shown in Fig. 7. (3) The optimum pumping rate is obtained when the vertical distance between the cost line and any water return line is greatest. However, when these optimum pumping rates are of identical values, whether excluding or including the well costs. It is evident that, when including the well costs, the unit price of water must be increased so that the water return covers the global cost. Analytical analysis of the slope of each water return line will give the new value of unit price of water which satisfies the maximum economic benefit, Table 6. Comparison between Fig. 6 and Fig. 7 shows that: (1)The new cost and WR lines in Fig. 7 have the same trends as those in Fig. 6. (2)Same values of optimum pumping rates are obtained at dif- ferent unit price of water in both cases. (3)The optimum Qt increases with an increase of unit price of water. (4)Including the well cost leads to use of greater unit price of water to make the income of water covers the total cost including well cost. 3.4.4 Including the Elevated Tank Cost In order to study the effect of elevated tank cost the same Table 5. Results of Optimum Pumping Rate by the Analytical and Graphical Solutions at Different Unit Price of Water (Opti- mum Commercial Sch. (3)) Water return line Unit price of water, b (LE/m3 ) Optimum pumping rate, Qt, (m3 /day) by the graphical solution Optimum pumping rate, Qt, (m3 /day) by the developed equation B1*Qt b = 0.100 6,000 6,880 B2*Qt b = 0.125 9,000 9,139 B3*Qt b = 0.150 10,000 10,880 B4*Qt b = 0.175 12,0000 12,362 B5*Qt b = 0.200 13,000 13,680 Fig. 7. Optimum Pumping Rate at Different Unit Price of Water Including the Well Cost, (opt. comm. Sch. (3)) Table 6. Effect of Excluding Initial Cost and Including the Well Cost on Optimum Pumping Rates at Different Unit Price of Water (Opt. Comm. Sch.(3)) Excluding Initial cost Including well cost Unit price of water, b (LE/m3 ) Water return B * Qt Optimum pumping rate, Qt (m3 /day) Water return B* Qt + Cwe Optimum pumping rate, Qt (m3 /day) New unit price of water, b' 0.100 273.75 * Qt 6,000 273.75*Qt +160000 6,000 0.109 0.125 342.187 * Qt 9,000 342.18*Qt +160000 9,000 0.131 0.150 410.625 * Qt 10,000 410.62*Qt +160000 10,000 0.156 0.175 479.06 * Qt 12,000 479.06 *Qt+160000 12,000 0.180 0.200 547.5 * Qt 13,000 547.5 *Qt +160000 13,000 0.204 Table 7. Effect of Excluding Initial Cost and Including the Elevated Tank Cost on Optimum Pumping Rates at Different Unit Price of Water (Opt. Comm. Sch.(3)) Excluding Initial cost Including elevated tank cost Unit price of water, b (LE/m3 ) Water return B*Qt Optimum pumping rate, Qt (m3 /day) Water return B* Qt + Cet Optimum pumping rate, Qt (m3 /day) New unit Price of water, b' 0.100 273.75 * Qt 6,000 273.75*Qt +536011.9 6,000 0.133 0.125 342.187 * Qt 9,000 342.18*Qt +536011.9 9,000 0.147 0.150 410.625 * Qt 10,000 410.62*Qt +536011.9 10,000 0.170 0.175 479.06 * Qt 12,000 479.06*Qt +536011.9 12,000 0.191 0.200 547.5 * Qt 13,000 547.5 *Qt +536011.9 13,000 0.215
- 7. Optimizing Pumping Rate in Pipe Networks Supplied by Groundwater Sources Vol. 17, No. 5 / July 2013 − 1185 − procedure and graphical solution are followed, Fig. 8. The results are given in Table 7. Comparing Fig. 6 and Fig. 8, shows similar results as in item (4.2). In addition, the increase of unit price of water in this case is greater than including solely the well cost, since the elevated tank cost is more than well costs. 3.4.5 Including the Initial Cost of Both Well and Elevated Tank In this case, the graphical solution is shown in Fig. 9, and the results are given in Table 8. The comparison between Fig. 6 and Fig. 9 shows the same results obtained here as well. In addition, the increase of unit price of water in this case is greater than when including either the well cost or elevated tank cost. This increase is being noticeable. 3.5 Economic Considerations 3.5.1 Pricing of Water In developing countries like Egypt, governments are faced with limited financial resources needed to provide funds for capital investment in water supply systems. Therefore, it is necessary to recover all or part of the investment, interest, operation, maintenance and other expenses in water-supply projects. Charging money for the delivery of water is a traditional way to promote this objective (Darwich, 1982). However, pricing of water for management projects is one of the measures to gain efficiency, productivity and revenue for the national economy. Water pricing should be implemented ensuring the price does not exceed its value to the user and/or his willingness to pay. 3.5.2 Results of Water Income Excluding the Initial Cost A summary of the results of the economic water income when excluding initial cost is presented in Fig. 10 which shows the relation between the calculated economic water income and the different pumping flow rates Qt at different unit price of water b, which ranging from 0.1 to 0.2 LE/m3 of water. Comparison between Fig. 6 and Fig. 10 shows that: 1. The water income increases with increase of unit price of water. 2. The b = 0.1LE/m3 line, shows a negative economic income at Qt more than 8500 m3 /day indicating a loss in water income, since WR is less than total cost. 3. In each line, the water income increases with increase of Qt, until a particular point at which the water return, WR, begins to drop. This is a turning point, at which the water income is greatest, and the Qt value is the optimum pumping rate at the corresponding b value. 4. The optimum Qt values indicated by arrows in Fig. 10 are identical with those shown in Fig. 6 and those interpreted from Table 4 for optimized commercial Sch. (3), indicating the validity of the developed analytical solution and the employed graphical solutions. Fig. 8. Optimum Pumping Rate at Different Unit Price of Water Including the Elevated Tank Cost (Opt. Comm. Sch. (3)) Table 8 Effect of Excluding Initial Cost and Including it on the Optimum Pumping Rates at Different Unit Price of Water (Optimum Com- mercial Sch.(3)) Excluding initial cost Including initial cost Unit price of water, b (LE/m3 ) Water return B*Qt Optimum pumping rate, Qt (m3 /day) Water return B* Qt + Cet+Cwe Optimum pumping rate, Qt (m3 /day) New unit price of water, b' 0.100 273.75 * Qt 6,000 273.75*Qt +696011.9 6,000 0.142 0.125 342.19 * Qt 9,000 342.18*Qt +696011.9 9,000 0.153 0.150 410.63 * Qt 10,000 410.62*Qt +696011.9 10,000 0.175 0.175 479.06 * Qt 12,000 479.06*Qt +696011.9 12,000 0.196 0.200 547.50 * Qt 13,000 547.50*Qt +696011.9 13,000 0.220 Fig. 9. Optimum Pumping Rate at Different Unit Price of Water Including Initial Cost (Optimum Commercial Sch. (3))
- 8. Mohamed M. Somaida, Medhat M. H. El-Zahar, Yasser A. Hamed, and Mahmoud S. Sharaan − 1186 − KSCE Journal of Civil Engineering 3.5.3 Water Income Results Including the Initial Cost The summary for the results of water income when including initial cost are presented in Fig. 11 and shows the relation between the calculated water income and the different pumping flow rates Qt at the different unit price of water b(which ranges from 0.14 to 0.22 LE/m3 of water. Comparison between Fig. 9 and Fig. 11 shows that: 1. The water income increases with increase of unit price of water. 2. In each line, the water income increases with increase of Qt, until a turning point where water income begins to drop slightly indicating the optimum Qt at the particular unit price of water. 3. The optimum Qt values shown by arrows in Fig. 11 are iden- tical with those shown in Fig. 9, since both consider the ini- tial cost indicating the validity of the analytical and graphical solutions. 4. The b values must be increased such that WR must cover the overall cost to maintain the same optimum pumping rate. This is evident in Fig. 9 and Fig. 11. 4. Conclusions An analytical solution has been reached to calculate the optimum pumping in a pipe network supplied from a groundwater source by means of water wells. The solution depends on; the unit economic return of water, total unit cost of water production, operating head of pump, well losses, operating, maintenance and labor costs. Application of the analytical solution to an example water distribution system determines values for optimum pumping rate which are in good agreement with those produced graphically. Graphical study of the effect of initial cost on the optimum pumping rate, reflects the importance of including the initial cost in the evaluation of the optimum pumping rate because of its bearing on the choice of the proper unit price of produced water, where it must be increased such that water return covers the global cost and hence the maximum economic benefit could be satisfied. The effect of unit price of water on the water return at different pumping rates has been also studied. It is found that, whether excluding or including initial cost, the water return increases with increase of of pumping rate till a turning point, after which the water return begins to decline slightly. Hence, at this point, the optimum pumping rate in the pipe network is rejected as well as the maximum economic benefit satisfied. Finally, within the scope of the present study, it is formed that the identity or good agreement shown in the results of optimum pumping rate, indicates the validity of the developed analytical solution and the graphical solution employed. It is recommended that: 1. The option of pricing water can be considered, since it is one of the measures to gain efficiency, productivity and income for the national economy. However, the price should not exceed its value to the user and/or his willingness to pay. 2. The pumping rates utilized in the P.N.W. must not exceed the safe discharge from wells for fear of deterioration of the groundwater source. Notations a= Pipe cost coefficient A1 = Unit cost of pumping A2 = Unit cost of pump b= Unit price of produced water B= Unit water return C= Cost coefficient for concrete elevated tank C1 = Cost coefficient of labor and maintenance C2 = Treatment cost coefficient C3 = Inclusive cost coefficient Cet = Elevated tank cost Cf = Formation loss coefficient Cin = Inclusive cost Clm = Labor and maintenance cost Cp1 = Pumping cost Cp2 = Pump cost Cpi2 = Pipe network cost CT = Total cost Ctr = Treatment cost Cw = Well loss coefficient Cwe = Well costs D= Diameter of pipe Fig. 10. Economic Water Income (LE) and Different Pumping Flow Rates Qt at Different Unit Price of Water, B (LE/m3 ), Exclud- ing Initial Cost (Optimum Commercial Sch. (3)) Fig. 11. Economic Water Income (LE) and Different Pumping Flow Rates Qt at New Unit Price of Water, B' (LE/m3 ), Including Initial Cost (Optimum Commercial Sch. (3))
- 9. Optimizing Pumping Rate in Pipe Networks Supplied by Groundwater Sources Vol. 17, No. 5 / July 2013 − 1187 − fi = Pipe friction factor Hd= Operating head of pump hfmax = Maximum head lost by friction encountered in the pipe network hL = Friction and minor losses in the main hr = Residual pressure head in the pipe network K/d = Relative roughness k = Loss coefficient in the main K= Pipe roughness height L = Length of pipe Li = Individual pipe length m = Number of pipes in network Q= Discharge Qi = Individual pipe discharge Qt = Daily pumping rate Qt = Pumping rate in P.N.W. Qw = Discharge per well r= Percent daily of pumping S= Hydraulic gradient Sw = Drawdown in the well WR = Water return Y= Life period of Scheme Z = Depth of water table from ground level before pumping References Bear, J. (1979). Hydraulics of groundwater, McGraw-Hill, New York, pp.1-81. Bouwer, H. (1978). Groundwater hydrology, McGraw-Hill, New York. Calborn, B. J. and Rainwater, K. A. (1991). “Well-field management for energy efficiency.” Journal of Hydrology Engineering, ASCE, Vol. 117, No. 10, pp. 1290-1303. Darwich, M. R. (1982). Proposed methods for water pricing and charging in egypt, MSc Thesis, International Center for Advanced Mediterranean Agronomic Studies, Bari, Italy. Egyptian Code of Practice and Specification (101-1997) for designed water and sewerage, Egypt. Featherstone, R. E. and El-Jumaily, K. K. (1983). “Optimal diameter selection for pipe networks.” Journal of Hydrology Engineering, ASCE, Vol. 109, No. 2, pp. 221-234. Fetter, C. W. (1994). Applied hydrogeology, Prentice Hall, New York. Naggar, O. M. (2002). “Groundwater production cost.” Proceedings of the International Water Technology Conference, Cairo, Egypt. Naggar, O. M. (2005). “Optimum design and construction of water wells.” 9th Proceedings of the International Water Technology Conference, Sharm El-Sheikh, Egypt. Rajput, R. K. (1998). A textbook of fluid mechanics, Rajendra Ravindra, New Delhi, India. Somaida, M. M. (1991). “Analytical solutions for determining economically optimum well discharge.” Port Said, Science Engineering Bulletin, Vol. 3, No. 1, pp. 16-24. Somaida, M. M. (1993). “Effects of the hydraulic and cost parameters on the economically optimum well discharge.” Port Said, Science Engineering Bulletin, Vol. 5, No. 1, pp. 25-34. Somaida, M. M., Elzahar, M. M., and Sharaan, M. S. (2011). “The use of computer simulation and analytical solutions for optimal design of pipe networks supplied from a pumped groundwater source.” Port-Said Engineering Journal, August, 2011 (Accepted to be published). Suez Canal University, Faculty of Engineering (2005). Hydraulic project, Civil Engineering Department, Suez Canal University, Ismailia, Egypt. Tung, Y. K. (1986). “Ground water management by chance-constrained model.” J. Water Resources Planning and Management, ASCE, Vol. 112, No. 1, pp. 1-19. www.epa.gov/ORD/NRMRL/wswrd/epanet.html (Rossman 1994, www. epa.gov).