Applying Solver Engine in Microsoft Excel to Analyze Water Distribution Network based on Linear Theory Method

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1 J. Saf. Environ. Health Res. 1(2): XX XX, Winter 2017 DOI: /jsehr ** ORIGINAL RESEARCH PAPER Applying Solver Engine in Microsoft Excel to Analyze Water Distribution Network based on Linear Theory Method Mohammad Khazaei 1 * 1 Research Center for Environmental Pollutants, Department of Environmental Health Engineering, Qom University of Medical Sciences, Qom. ABSTRACT: The linear theory method (LTM) equipped with Solver engine in Microsoft Excel was applied to analyze the water network. The linear theory method was served as a mathematical tool to convert non-linear equations derived from head lose balance in loops to linearized forms. Tables including sets of pipes, nodes, and loops were arranged in an Excel datasheet. The appropriate equations assigning to discharge continuity in nodes and head loss balance in loops were constructed. Equations were solved by applying Solver engine and pipe discharges were the unknowns. The differences between obtained pipe discharges in two consecutive iterations (ΔQ) were considered as an indicator to identify the calculation trend. Approaching the ΔQ values to zero during the iterations indicates progressing the calculation to an appropriate convergence. A simple command was provided via Visual Basic language for replacing the data during the iterations. This method solves whole network equations, simultaneously and therefore it is more efficient than Hardy-Cross method which only solves one equation at a time. Also, the linear theory method uses the simplicity and uniform nature of Darcy-Weisbach or Hazen-Williams formulas obtaining in water network systems and avoids sophisticated calculations like those applied in Newton-Raphson method. KEYWORDS: Linear theory method, solver add-in, water distribution system, Hardy-Cross, Newton-Raphson. Introduction Access to safe drinking-water is vital to human health, and as wall as a basic component of effective health protection [1, 2]. The water distribution networks are critical component of every drinking water supply system. The principal goal of pipe networks is providing the appropriate quantity and quality of water [3]. Networks based on loop method allow supplying water in a high level of quality and in the case of occurring accidents on the network, the downstream zones can be still remain in service [4]. At least some equations involved in pipe network analysis are non-linear. Because no direct method is available for their solution, these equations should be linearized and then can be solved via a trial method [5, 6]. Several methods have proposed to solve the equations of loop networks such as Hardy-Cross, Newton-Raphson, and linear theory method. Both, Hardy-Cross and Newton-Raphson methods are based on Taylor series which converts the non-linear forms of equations to the linear forms [7]. In Hardy-Cross method, a ΔQ value is considered for each loop and after several iterations, the amount of ΔQ approaches to the zero. The main drawback of Hardy-Cross method is solving only one equation at a time and ignoring the effect of neighbor loops [8]. Some corrections have been done on the Hardy-Cross method which have a little effect on improving its efficacy [9]. The application of the Newton-Raphson method to the analysis of water distribution networks was first proposed by Martin and Peters [10, 11] and since then it has been ex- * Corresponding Author khazaei@muq.ac.ir Tel.: ; Fax: Note. Discussion period for this manuscript open until Jaly 31, 2017 on JSEHR website at the Show Article tensively used in research and in the field. Newton-Raphson method solves all equations of the network, simultaneously. This method can be applied for both ΔH and ΔQ equations. The main concept is based on Newton-Raphson algorithm derived from Taylor series which calculates the x 1 value according to x 0 supposed amounts as follows [12]: Forming complicated matrix comprised different derivations and solving the matrix with several iterations are the main problems of Newton-Raphson method [7]. The linear theory method emphasizes the nature of equations applied in water distribution systems which are originated from Darcy-Weisbach or Hazen-Williams formulas [13]. Darcy-Weisbach and Hazen-Williams formulas have non-linear forms and the power of discharge quantity (Q) are 2 and 1.852, respectively [14]. The simple numbers 2 and as Q power can be altered to the linear forms without applying the Jacobin method [7, 13]. But, solving a complicated matrix comprises numerous linear and linearized equations is still a problem. Solver is an add-in tool available in Microsoft Excel. Excel Solver is an efficient tool for optimization targets. The Solver is a combination of a various programs, modeling language, different optimizers to solve linear, and non-linear equations having a simple Graphical User Interface (GUI). With the combination of these powerful tools, Solver has the capability to obtain the ideal values in a worksheet which is considered for target cell [15, 16]. In this work, the linear theory method was applied to analyze the water piped network. The obtained matrix of linear and linearized equations was formed and Solver engine in Microsoft Excel was used to find the optimum pipe discharge amounts.

2 Materials and methods Basic concepts The aim of solving the network is to determine the discharges of pipes connected to the nodes (junctures). So, the number of unknown quantities (pipes discharge, Q x ) is equal to the number of pips, X. Following relationship between pipes, nodes, and loops is applicable in a loop pattern network: According to the linear theory method, the unknown term, Q n, is considered as two parts including a non-linear term, Q n-1, which is merged with pipe resistance constant (R) to form R and a linear part, Q 1, as follows: Where, X, J, and C are the number of pipes, nodes, and loops, respectively, which construct network. To solve a matrix consisting of X unknown quantities, we need at least X equations. There are two basic assumptions in loop networks: Assumption 1: All inlet discharges into a node are equal to the outlet discharges which flow out of the node (Eq. 3). The supposed value of Q in non-linear term, Q n-1, is 1 for the first iteration. Consequently, the non-linear equation is linearized in the form of following equation: Where, Q x is discharge rate of a pipe connected to a node (juncture). If the flow direction is into the node, the sign of value Q x is positive and if the flow direction is out of the node, the value of Q x should have the negative sign. q j is the discharge demand of a node which can be identified as water consumption rate (m 3 /s). Assumption 2: The sum of head loses (h l ) in pipes belonged to a certain loop (assigning to the clockwise or non-clockwise direction of the flow) is zero as represents in Eq. 4. Where, h l is head loss (m), x and c are the numbers of pipes and loops, respectively. By applying assumption 1, J equations can be constructed. Also, C equations can be formed by using assumption 2. Consequently, J +C equations are formed which have +1 equation more than X unknown quantities according to Eq. 2. So, the number of equations is satisfied and the constructed matrix can be solved. J equations formed according to assumption 1 are simple linear relationships. C equations formed based on assumption 2 can be extended by Darcy-Weisbach or Hazen-Williams equations as follows: As represented in Eq. 7, C linearized equations are formed. Considering J linear equations which were formed according to Eq. 3, the number of linear equations (J +C) is satisfied to establish a linear matrix. Results Test network Fig. 1 shows a one-source, eight-demand node network. The number of nodes, pipes, and loops has denoted in Fig. 1. Fig. 2 represents the consecutive steps to solve the equations for determining the unknown values, Q x. Where, R is the constant of pipe resistance that has different formula if be considered in Darcy-Weisbach or Hazen-Williams equations. n is the power of unknown quantity, Q x, which has the values 2 and in Darcy-Weisbach and Hazen-Williams equations, respectively. Linear theory method Fig. 1. A schematic map of a one-source, eight-demand node network. (Source/demand discharge of nodes, initial discharge of pipes, and length of pipes are identified on the map by using [q j as m 3 /s], (Q i as m 3 /s), and [17], respectively). As can be observed in Eq. 5, the unknown quantity, Q x, obtains power n 1, so that the equation has non-linear form and it cannot be solved in a linear matrix.

Flowchart of steps required to construct the appropriate nodal and loop equations for

3 Mohammad Khazaei et al. / J. Saf. Environ. Health Res. 1(2): XX XX, Winter 2017 Fig. 2. Flowchart of steps required to construct the appropriate nodal and loop equations for applying the solver engine. Fig. 3. A Microsoft Excel datasheet containing tables of pipes, nods, and loops. The cell address of important parameters was provided in the inset box.

Fig. 4. A Solver engine window and the setting of its options applied in this work. Table 1. Pipe discharges and ΔQs obtained from consecutive iterations using Solver engine.

4 Fig. 4. A Solver engine window and the setting of its options applied in this work. Table 1. Pipe discharges and ΔQs obtained from consecutive iterations using Solver engine. Pipe Iteration 1 Iteration 2 Iteration 3 Iteration Iteration 12 Assumed Obtained Assumed Obtained Assumed Obtained Assumed Obtained Assumed Obtained ΔQ x 1 ΔQ x 2 ΔQ x 3 ΔQ x ΔQ x The network has 12 pipes and therefore the basic Q unknowns are Q 1... Q 12. According to the node-flow continuity relationships at nodes (see Fig. 1), the following 9 linear equations are produced:

5 The linear theory method by assisting Solver engine in Microthe pipes are obtained as represented in Table 2. Applying these R values, the head loss equations for loops I... IV are respectively, Taking f= for all pipes, the first iteration R values of Table 2. Values of pipe discharge (Q), Darcy friction coefficient (f), and pipe resistance (R and R ) considered or obtained in the first and second iterations. Iteration 1 Iteration 2 Pipe Q 1 x f 1 x R 1 x 1 R X Q x(1) 2 Q x 2 f x 2 R x 2 R X Q x(2) Considering 1 Q 1 =... = 1 Q 12 = 1 for the first iteration, linearization forms of the equations (17)... (20) are performed in Table 3. And the linearized equations (25)... (28) are constructed. Equations (8)... (16), and (25)... (28) are solved simultaneously and the Q 1(1)... Q 1(12) values are obtained by applying Solver engine as illustrated in Figs. 4 and 5. Using the assumed Q values, the f and R values are updated, and Equations (17)... (20) are linearized and solved simultaneously along with equations (16)... (25). The iterative procedure is continued and the first four iteration results and 12 th iteration result are shown in Table 3. The assumed values for iteration 12 would be practically the final answer because the difference between two consecutive iteration results is negligible (ΔQ x <0.001 L/s). Fig. 5 shows the decreasing trends of ΔQ x during 40 iterations by Solver engine which reveals increasing the closeness of consecutive pairs of Q x values. As observed in Fig. 3, a button namely, "column change" has been provided in the Excel datasheet to facile date moving from column Q x to column Q x between consecutive iterations. It is supported by Visual Basic language that is available in Developer tab of Excel software and it works according to the following command; Sub Macro3() Macro3 Macro F Keyboard Shortcut: Ctrl+Shift+G Range("E3:E11").Select Selection.Copy Range("D3:D11").Select Selection. Paste Special Paste:=xlPaste Values, Operation:=xlNone, Skip Blanks _ :=False, Transpose:=False End Sub Discussion Table 3. The linearized forms of equations (17)... (20) by applying 1 Q x = 1. Equation No. (21) (22) (23) (24)

6 Fig. 5. Decreasing trends of ΔQ x values as the iteration numbers are increased. soft Excel was applied to find the unknown quantities of pipe discharge (Q x ) in a water loop network. Comparing with the Hardy-Cross method that is relied on solving the equations of each loop distinctly, the linear theory method can solve the equations of whole network, simultaneously. Hardy-Cross method assumes only one equation at a time from the available set of ΔQ equations, neglects the effect of adjacent loops, and ΔQ powers higher than 1, which are expanded by Taylor's Series, are omitted. The effect of neglecting higher-power correction terms in Hardy-Cross method is tolerable. However, the effect of neglecting adjacent loops and considering only one correction equation at a time is considerable. Both the Hardy-Cross method and Newton-Raphson method comprise the equations expanded by Taylor's Series. The Newton-Raphson method is based on equation 1 and is used to find the real roots of f(x)= 0 if f'(x) is a simple expression and easily found. The NRM expands the non-linear terms in Taylor's series, neglects the residues after two terms, and thereby considers only the linear terms. As such, the Newton-Raphson method linearizes the non-linear equations through partial differentiation. This method is general and works even when the non-linear equations are transcendental containing exponential, trigonometric, hyperbolic, or logarithmic terms. The non-linearity in the equations for pipe network analysis, however, is algebraic, uniform and simple: the variables are raised to the same, non-unity exponent. For example, the non-linear Q equations obtained from the loop-head loss relationship contain terms, R x Q n. The value of n is the same for all terms (1.852 in the Hazen-Williams pipe head loss formula and 2.0 in the Darcy-Weisbach and Manning formulas). This is a helpful feature and the non-linear term can be easily linearized by merging a part of the non-linear term into the pipe resistance constant as in equation 6. This principle was first suggested and used by Mcllroy [18], Marlow [19], and Muir [20]. Later, Wood and Charles [21] developed it, and it is now widely used in practice [21-23]. The amount of ΔQ, which is the difference between Q x values, is obtained from two consecutive iterations. The ΔQ used in this work differs from those applied in HCM and it serves as an indicator to judge about satisfying the iterations in LTM. As shown in Table 5, more than 10 iterations are required to obtain the satisfied Q x values. Requiring to perform the numerous iterations is the main problem attributed to LTM, especially in pipe networks having considerable number of loops. Applying solver engine is an efficient approach to solve all linear and linearized equations, simultaneously. The proposed method applies the LTM for producing linear equations, the Solver engine to perform iterations, the Visual Basic command to fast data moving, and the ΔQ indicator to identify when the Q x values are satisfied. Conclusion A facile approach relying on the linear theory method, which is enabled by Solver engine in Microsoft excel software, was applied to analyze pipe water network. The following features of this experiment can be mentioned: Applying the linear theory method to convert non-linear equations to linearized equations. Using Solver engine to analyze the matrix comprising the linear and linearized equations. Replacing Q x values, obtained from each iteration, by Q x (n- 1) unknowns that was merged with R values according to the linear theory method. Facile replacing the pipe discharge data during consecutive iterations by using a simple Visual Basic command. Using all equations derived from nodes and loops. Consequently, the number of equations are exceeded the number of unknowns. This helps to occur a faster convergence. Proposing ΔQ value as an indicator to identify the progressive trend of iterations and a tool to judge the precision of results. A mono-source pipe network was proposed as example in this experiment. However, the networks containing various features such as pumps, valves, and several water sources can be applied. Acknowledgments I would like to thank my friend Dr. Reza Fouladi Fard, an assistant professor in Qom University of Medical Sciences, for his valuable advises in Visual Basic language and my students for

7 solving their own projects based on this approach to verify its consistency. References [1] WHO, Guidelines for Drinking-Water Quality, World Health Organization, Geneva, [2] T.S. Yekta, M. Khazaei, R. Nabizadeh, A.H. Mahvi, S. Nasseri, A.R. Yari, Hierarchical distance-based fuzzy approach to evaluate urban water supply systems in a semi-arid region, Journal of Environmental Health Science and Engineering, 13 (1)(2015) 53. [3] NRC, Drinking Water Distribution Systems: Assessing and Reducing Risks, National Academies Press, [4] R. Shukla, A feasibility analysis of the use of the Skipworth model for Bridgetown Water Supply Network and a Whole Life Costing to compare the water distribution systems at Greenvale pumping station, (2012). [5] P.K. Swamee, A.K. Sharma, Design of Water Supply Pipe Networks, John Wiley & Sons, [6] M. Collins, L. Cooper, R. Helgason, J. Kennington, L. LeBlanc, Solving the pipe network analysis problem using optimization techniques, Management Science, 24 (7)(1978) [7] P.R. Bhave, Analysis of Flow in Water Distribution Networks, in: Analysis of Flow in Water Distribution Networks, Technomic Publishing, [8] T.J. McGhee, E.W. Steel, Water Supply and Sewerage, McGraw-Hill New York, [9] N. Moosavian, M.R. Jaefarzadeh, Hydraulic analysis of water supply networks using a Modified Hardy Cross method, International Journal of Engineering, Transactions C, 27 (9)(2014) [10] G. Farina, E. Creaco, M. Franchini, Using EPANET for modelling water distribution systems with users along the pipes, Civil Engineering and Environmental Systems, 31 (1)(2014) [11] M.M. Vaseti, Simulate the water distribution networks by improved newton-rapson method, International Journal of Academic Research, 3 (3)(2011). [12] L.J. Goldstein, D.C. Lay, D.I. Schneider, N.H. Asmar, Calculus & its Applications, Pearson Higher Ed, [13] P.R. Bhave, Optimal Design of Water Distribution Networks, Alpha Science Int'l Ltd., [14] R.V. Giles, J.B. Evett, C. Liu, Schaum s Outline of Fluid Mechanics and Hydraulics, McGraw-Hill, [15] D. Fylstra, L. Lasdon, J. Watson, A. Waren, Design and use of the Microsoft Excel Solver, Interfaces, 28 (5)(1998) [16] I. Griva, S.G. Nash, A. Sofer, Linear and Nonlinear Optimization, Siam, [17] Guidelines for Drinking-water Quality, WHO Chronicle, 38 (2011) [18] M.S. McIlroy, Pipeline network flow analysis using ordinary algebra, Journal (American Water Works Association), 41 (5)(1949) [19] T.A. Marlow, R.L. Hardison, H. Jacobson, G.E. Biggs, Improved design of fluid networks with computers, Journal of the Hydraulics Division, 92 (4)(1966) [20] J. Muir, Discussion of "Improved Design of Fluid Networks with Computers," by TA Marlow et al, Journal of Hydraulic Division, ASCE, 93 (HY2)(1967) [21] D.J. Wood, C.O. Charles, Hydraulic network analysis using linear theory, Journal of the Hydraulics Division, 98 (7)(1972) [22] A.G. Collins, R.L. Johnson, Finite-Element method for water-distribution networks, Journal (American Water Works Association), (1975) [23] L.T. Isaacs, K.G. Mills, Linear theory methods for pipe network analysis, Journal of the Hydraulics Division, 106 (7)(1980) AUTHOR(S) BIOSKETCHES Khazaei, M., Ph.D., Assistant Professor, Faculty of Health, Department of Environmental Health Engineering, Qom University of Medical Sciences, Qom, Iran. khazaei@muq.ac.ir COPYRIGHTS copyright for this article is retained by the author(s), with publication rights granted to the journal. this is an open access article distributed under the terms and conditions of the Creative Commons Attribiotion Licsense ( HOW TO CITE THIS ARTICLE G.H. Safari, S. nasseri, A.H. Mahvi, M. Hoseini, H. Kamani, S.D. Ashrafi, Environmental Monitoring of Tetracycline Antibiotic in the Aquatic Environments in Mianeh, Iran, Journal of Safety, Environment, and Health Research, 1(2)(2017) XX XX. DOI: /jsehr ** URL:

Appendix F. + 1 Ma 1. 2 Ma Ma Ma ln + K = 0 (4-173)

Appendix F. + 1 Ma 1. 2 Ma Ma Ma ln + K = 0 (4-173) 5:39p.m. Page:949 Trimsize:8.5in 11in Appendix F F.1 MICROSOFT EXCEL SOLVER FOR NON-LINEAR EQUATIONS The Solver is an optimization package that finds a maximum, minimum, or specified value of a target