# Numerical and Theoretical Investigation to Estimate Darcy Friction Factor in Water Network Problem Based on Modified Chun-Hui He’s Algorithm and Applications

Authors: Waseem Asghar Khan; Muhammad Arif; Mogtaba Mohammed; Umer Farooq; Fozia Bashir Farooq; Murtada K. Elbashir; Jamshaid Ul Rahman; Ziyad Ali AlHussain
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this work, we applied the modified Chun-Hui He’s algorithm to evaluate for estimation of flow friction factorf for value of friction factor by using Colebrook–White relation. The speedy, precise, and consistent evaluation of flow friction factor f are essential for evaluation of pressure dips and streams in complex network prototypes at distinct values of diameters of pipes. Friction factor estimated outcomes are applied in everyday engineering routine. Numerous computational systems tested for distinguishing of water pipe networks resolution, such as Hardy Cross method (HCM), Newton method (NM), and modified Newton method (MNM), are presented. As a novelty, a modified Newton method tabulated data, graphical results, and comparisons that are presented with different numerical schemes.


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## Body

## 1. Introduction

The investigation of a few issues in many fields such as computational physics, computational biology, engineering, environmental sciences, chemistry, and economics in order to resolve real-life nonlinear models with constrained domain. The Newton technique and its variations are effective in solving nonlinear models that occur in real-world problems with reasonable stopping conditions [1, 2]. The flow variation to all loops is immediately removed in this manner, resulting in a high-level merger. This technique nevertheless requires suitable essential assumptions for tributary levels that sustain progression circumstances and are adjacent to the specific stream [3, 4].The stream systems are energetic, and complex frameworks involve gigantic ventures by private and government sectors. These sectors need sufficient management to control professionally to accomplish goals of their system [5–11]. Unfortunately, now a days, the management of water resource systems is challenging and problematic due to the rapidly increasing customer requirements. This situation is difficult and challenging for conventional methodologies to manage these circumstances [12–14]. Recently, computational methodologies have been tried to tackle these complex circumstances [11, 12]. To explore hydraulic movement in complicated network systems and to satisfy the energy and continuity equations, these quantitative approaches use the Hardy Cross method. Various research laboratories have recently used various techniques to solve these limitations, such as sluggish convergence and recurring dissatisfaction with outcomes, and have failed to meet consumer demand criteria. The failure of others’ attempts to eliminate all problems sparked the breakthrough. Furthermore, the Hardy Cross method must be modified due to the nonlinear nature of tube systems [5, 15, 16].In electrical systems, the relationship between voltage and current with regular resistors is governed by Ohm’s law with diodes where resistances depending on current and voltage are nonlinear electrical circuits containing nonlinear components and solving second-order coupled nonlinear Schrödinger equations by using various numerical approaches [3, 4]. HCM is followed by other methods that adopted the Newton–Raphson method, one of the faster effective procedures with higher convergence [17–19]. Toldini and Pilati [20] proposed a global gradient method that is created as a variation to the NM. Such methodologies are usually applied to resolve a system of nonlinear algebraic equations that communicate the behaviors of the hydraulic structures [21–23]. To solve nonlinear problems, numerical approaches are a reasonable alternative. The well-known numerical approaches are visible in [3].In this study, we believe in a new estimated scheme that is a variation of the standard NM, and we evaluated its efficacy alongside the NM and HCM. AtRe≥4000, the modified Chun-Hui He’s algorithm [24] was used to calculate the friction factor for given pipe diameters in turbulent flow in a confined region, where Re is called Reynolds number and ε=0.05 is roughness height [25, 26]. At the next level, we used fraction factor values to obtain the water pressure function for various lengths, diameters, and water flow rates in each pipe. All computational data are first incorporated into an Excel sheet, and then, Mathematica code is used to describe the consequences of network systems using Excel data [6].


## 2. Structure Topology of the Hydraulic System

The first step in characterizing a hydraulic problem is to create a network composition that shows pipe connections in terms of diameter, length, and nodes. Water and utilization levels from suppliers should be allocated to intersection points. Instead of using availability to track pipes, metrics are assigned to each tube and the system’s closed loop, as shown in Figure1. The next step demonstrates the pipes system for preliminary supply of stream is used for utilization in every intersection point and should obey Kirchhoff’s law [27]. The overall water entering at an intersection point is approximately the same of that leaves that enter intersection point of the network. The similar preservation law is to satisfy the entire system.Figure 1
The problem with the water supply network prompted by Kirchhoff’s law [3].


## 3. Topology of the Hydraulic Model

A scientific explanation of the model can be developed once a complex system configuration is created, along with its loop numbers, pipeline, supply, and resource data. According to the mineralogy theorem of Euler, M nodes (intersection points) and N branches make up the system. In above network problem, we haveN=13 and M=10 and M−1 autonomous intersection points, i.e., 9 points and other one point is known as referent point and N−M+1=13−10+1=4 autonomous loops. In over problem, point J is called referent node.The Darcy–Weisbach equation and Colebrook–White relation for the Darcy friction factorf can be used to analyze this pipe network [8, 28].(1)Ww≈Δw=w1−w2=8fLq2gD5π2,where Ww is the water pressure function. L=lim,D=dimandq=qim3/sfori=1,2,…,13 are the pipes length, diameter, and flow vectors, respectively, and g is the gravity (m/s2).Taking the first derivative of equation (1) and q assumed as a variable herewith, we have(2)Ww′≈∂Wwq∂q=16fLqgD5π2.Darcy friction factorf can be described as(3)1f=−2log10ε3.71diameter+2.51Re.f.Colebrook–White (3) is applicable only at turbulent flow in restricted domain at Re≥4000, where Re is called Reynolds number and ε=0.05 is roughness height [25, 26, 28].


### 3.1. Modified Chun-Hui He’s Algorithm for Friction Factorf

The quick, precise, and dependable evaluation of friction factorf in (3) are essential for estimation of pressure falls in complex network models [6, 29]. So, we utilize the modified Chun-Hui He’s algorithm [24] to solve the approximate value of friction factor f from equation (3), and we found that the modified Chun-Hui He’s algorithm is up-to-date, and an efficient algorithm for solving nonlinear equations exists in real-life applications.(3) can be redeveloped as follows:(4)Ff,εD,Re=1f+2log10ε3.71Diameter+2.51Re.f=0.The friction factorf is used as a variable in this case, and we carefully select preliminary estimates to begin the numerical method in the given restricted domain.


#### 3.1.1. Modified Chun-Hui He’s Algorithm [24]

Step 1.
Necessary condition.(5)Fx0Fx1<0,where x0andx1 are the starting assumptions.Step 2.
Ancient Chinese algorithm computes(6)x2=x0−Fx0Rx0,x1,where Rx0,x1=Fx0−Fx1/x0−x1.Step 3.
Corrector step:(7)x3=x2−Fx2F′x2.Step 4.
Assumptions:(8)x0=x3−δ,x1=x3+δ,(9)δ=−Fx3F′x3.Step 5.
If we not get required accuracy, then go back to Step2.
Table1 provides the estimated fraction factor fn flow numbers using the modified Chun-Hui He’s Algorithm [24] for various pipe diameters (m).Table 1
The statistics values of flow, diameter, length, and constants value K displayed along with each value of the fraction factorf of each pipe in the network (Figure 2).
PipesNo.fnAssume flow (lt/s)Assume flow (m3/s)Diameter (m)Length (m)K = 8 fL/gD5п2A-B10.4470154801206153q1 = 125q1 = 0.1250.457120222.5804287B-C20.4470154801206153q2 = 75q2 = 0.0750.457150278.2255359C-D30.4470154801206153q3 = 35q3 = 0.0350.457150278.2255359D-G40.4470232660486057q4 = 25q4 = 0.0250.406160536.2707478G-J50.4470154801206153q5 = 90q5 = 0.0900.457120222.5804287J-I60.4470000072957691q6 = 150q6 = 0.1500.609300132.4056409I-H70.4470466694856617q7 = 175q7 = 01750.3041201708.9539397H-A80.4470466694856617q8 = 175q8 = 01750.3042803987.5591927B-E90.4470232660486057q9 = 50q9 = 0.0500.406160536.2707478C-F100.4470232660486057q10 = 40q10 = 0.0400.4061653.6270748E-F110.4470232660486057q11 = 15q11 = 0.0150.4061653.6270748F-G120.4470232660486057q12 = 15q12 = 0.0150.4061653.6270748E-I130.44703328769837464q13 = 25q13 = 0.0250.3551491.8100660Figure 2
Water supply network problem [3].


### 3.2. Loops Model in Water Network Topology

The system of nonlinear equations along each loop in this section was simulated.Loop I:WI,(10)Δw1+Δw9−Δw13−Δw7+Δw8=8gπ2f1l1q12d15+f9l9q92d95−f13l13q132d135−f7l7q72d75+f8l8q82d85,=222.5804287q12+536.2707478q92−91.8100660q132−1708.9539397q72+3987.5591927q82.Loop II:WII,(11)Δw2+Δw10−Δw11−Δw9=8gπ2f2l2q22d25+f10l10q102d105−f11l11q112d115−f9l9q92d95,=278.2255359q22+53.6270748q102−53.6270748q112−536.2707478q92.Loop III:WIII,(12)Δw3−Δw4+Δw12−Δw10=8gπ2f3l3q32d35−f4l4q42d45+f12l12q122d125−f10l10q102d105,=278.2255359q32−536.2707478q42+53.6270748q122−53.6270748q102.Loop IV:WIV,(13)−Δw5−Δw6+Δw13+Δw11−Δw12=8gπ2−f5l5q52d55−f6l6q62d65+f13l13q132d135+f11l11q112d115−f12l12q122d125.=−222.5804287q52−132.4056409q62+91.8100660q132+53.6270748q112−53.6270748q122.Above loop relations,WI,WII,WIII, andWIV can be written in matrix form:(14)100000−111000−101000000−11−100001−100000−10100000−1−100001−11×Δw1Δw2Δw3Δw4Δw5⋮Δw13=0,where l1,l2,…,l13 are the lengths of each pipe, d1,d2,…,d13 are the diameters, and water flow in each pipe, as shown in Figure 1, is q1,q2,…,q13, respectively. The positive sign showing flow direction is clockwise in left matrix of (14) and vice versa [5, 9, 30].


### 3.3. Nodes Model Topology

In this section, we will simulate the system of linear equation along each node by using Kirchhoff’s first law:(1)
Node A:−q1+q8−qA−output=0, where qA−output=50lt/s(2)
Node B:q1−q2−q9−qB−output=0, where qB−output=0(3)
Node C:q2−q3−q10−qC−output=0, where qC−output=0(4)
Node D:q3+q4−qD−output=0, where qD−output=60lt/s(5)
Node E:q9+q13−q11−qE−output=0, where qE−output=60lt/s(6)
Node F:q11+q10+q12−qF−output=0, where qF−output=70lt/s(7)
Node G:q5−q4−q12−qG−output=0, where qG−output=50lt/s(8)
Node H:−q7−q8+qH−input−qH−output=0, where qH−output=0,qH−input=350(9)
Node I:q7−q6−q13−qI−output=0, where qI−output=0(10)
Node J:q6−q5−qJ−output=0 where qJ−output=60lt/sand referent.Now, rewrite the above linear system of equations in matrix form [31]. The drawback of this matrix form is not a linear independent, that is why referent node would be omitted from the system of linear equation.(15)1000001000001−100000−1000001−100000−1000001100000000000000010−10100000000111000−11000000−1000000−1−1000000000−1100000−19×13×q1q2q3q4q5⋮q1313×1=qA−output=50qB−output=0qC−output=0qD−output=60qE−output=60qF−output=70qG−output=50qH−output−qH−input=−350qI−output=09×1.Above nodeJ count up as a referent, the left matrix in system (15) shows the rows corresponding to node A, node B, … node I, respectively.


### 3.4. Hardy Cross Method

The Hardy Cross method is applied to multiloop water network problems. Figures3 and 4 show the complete results of the HCM. This method converges to the required results after 24th iteration.Figure 3
The numerical results of head loss of equation (1) by using HCM and head loss in each loop going to zero after 24th iteration by selecting the random original flow in network problem.Figure 4
The numerical results of correction flow of equation (1) by using HCM and total corrected flow (m3/s) in each loop going to nil after 24th iteration with selecting random original flow in network problem, where ΔQI, ΔQII,ΔQIII, and ΔQIV change in flow of loop I–loop IV, respectively.The primary stage in resolution the problem is to make a net map display lengths and diameters and networks nodes. Later, we compose the original stream supply all through a pipe network. The selection of original flows must fulfill Kirchhoff’s law used in the system, and conservation law is also applicable for the entire network system [3, 13, 14, 27].Outcomes of HCM in column 7 indicate that after 24th iteration −Ww/Ww'(head loss) is nil by Darcy–Weisbach relation (1). The negative sign indicates anticlockwise water flows shown in Figure 1. Assign a sign with new flow of water. We will add in clockwise if ΔQ is +ve and vice versa. Similarly, we will subtract value if ΔQ is −ve and vice versa.


### 3.5. Newton Method

In this paragraph, hydrological problem is investigated by utilizing the NM having 2nd-order [32]. NM [23] or Newton-like procedure is not essential to take up an original estimate that fulfills the continuity law [33], as shown in Figure 1.Table2 provides the numerical results of correction flow of (1) by using conventional NM and change in two consecutive flows in each loop going to zero after 5th iteration by selecting the random original flow in network problem. The effectiveness of NM is better than HCM.Table 2
The water problem (Figure2) displaying the first iteration of technique NM ([32]). It is not necessary to follow Kirchhoff’s law in NM.
PipesLoopsDiameter (m)Length (m)qm3/sSign (q)Ww, equation (1)NM,qm3/sΔqm3/s10.4571200.12513.480.0892094−0.03579069I0.4061600.0511.340.0352368−0.0147632130.355120−0.025−1−0.490.1124880.08748870.304120−0.175−1−52.340.2107910.03579180.3042800.1751122.120.139209−0.035791LoopΣ74.110.036934220.4571500.07511.570.0539726−0.021027410II0.4061600.0410.860.017803−0.022197110.406150−0.015−1−0.110.08772490.072724990.406160−0.05−1−1.340.0352368−0.0147632LoopΣ0.970.014737330.4571500.03510.340.03616960.00116964III0.406160−0.025−1−0.340.0238304−0.0011696120.4061500.01510.11−0.0355278−0.0505278100.406160−0.04−1−0.860.017803−0.022197LoopΣ−0.74−0.072724850.457120−0.09−1−1.800.0383025−0.05169756IV0.609300−0.15−1−2.980.0983025−0.0516975130.3551200.02510.490.1124880.087488110.4061500.01510.110.08772490.0727249120.406150−0.015−1−0.11−0.0355278−0.0505278Σ−4.290.0062901All bold values are based on multidimension.


### 3.6. MNM (Modified Newton Method)

In this subsection, MNM achieves an optimum resolution of the nonlinear model simulated in Sections3.2 and 3.3 for scrutinizing the hydrological network considering. Present procedure is cost-effective and takes a smaller amount of time to attain the results than the HCM and conventional NM.Table3 provides the numerical results of correction flow of (1) by using MNM and change in two consecutive flows in each loop going to zero after three iterations by selecting the random original flow in network problem.Table 3
The first iteration of MNM ([34]) for the water problem (Figure 2).
PipesLoopsDiameter (m)Length (m)qm3/sSign (q)Ww, equation (1)MNM,qm3/sΔqm3/s10.4571200.12513.480.0879253−0.03707479I0.4061600.0511.340.0315287−0.0184713130.355120−0.025−1−0.490.1151210.09012170.304120−0.175−1−52.340.2120750.03707580.3042800.1751122.120.137925−0.037075LoopΣ74.110.03457520.4571500.07511.570.0563965−0.018603510II0.4061600.0410.860.0225996−0.0174004110.406150−0.015−1−0.110.086650.0716590.406160−0.05−1−1.340.0315287−0.0184713LoopΣ0.970.017174830.4571500.03510.340.0337969−0.00120314III0.406160−0.025−1−0.340.02620310.0012031120.4061500.01510.11−0.0392496−0.0542496100.406160−0.04−1−0.860.0225996−0.0174004LoopΣ−0.74−0.0716550.457120−0.09−1−1.800.0369535−0.05304656IV0.609300−0.15−1−2.980.0969535−0.0530465130.3551200.02510.490.115121−0.0134879110.4061500.01510.110.086650.07165120.406150−0.015−1−0.11−0.0392496−0.0542496Σ−4.29−0.1021805All bold values are based on multidimension.


## 3.1. Modified Chun-Hui He’s Algorithm for Friction Factorf

The quick, precise, and dependable evaluation of friction factorf in (3) are essential for estimation of pressure falls in complex network models [6, 29]. So, we utilize the modified Chun-Hui He’s algorithm [24] to solve the approximate value of friction factor f from equation (3), and we found that the modified Chun-Hui He’s algorithm is up-to-date, and an efficient algorithm for solving nonlinear equations exists in real-life applications.(3) can be redeveloped as follows:(4)Ff,εD,Re=1f+2log10ε3.71Diameter+2.51Re.f=0.The friction factorf is used as a variable in this case, and we carefully select preliminary estimates to begin the numerical method in the given restricted domain.


### 3.1.1. Modified Chun-Hui He’s Algorithm [24]

Step 1.
Necessary condition.(5)Fx0Fx1<0,where x0andx1 are the starting assumptions.Step 2.
Ancient Chinese algorithm computes(6)x2=x0−Fx0Rx0,x1,where Rx0,x1=Fx0−Fx1/x0−x1.Step 3.
Corrector step:(7)x3=x2−Fx2F′x2.Step 4.
Assumptions:(8)x0=x3−δ,x1=x3+δ,(9)δ=−Fx3F′x3.Step 5.
If we not get required accuracy, then go back to Step2.
Table1 provides the estimated fraction factor fn flow numbers using the modified Chun-Hui He’s Algorithm [24] for various pipe diameters (m).Table 1
The statistics values of flow, diameter, length, and constants value K displayed along with each value of the fraction factorf of each pipe in the network (Figure 2).
PipesNo.fnAssume flow (lt/s)Assume flow (m3/s)Diameter (m)Length (m)K = 8 fL/gD5п2A-B10.4470154801206153q1 = 125q1 = 0.1250.457120222.5804287B-C20.4470154801206153q2 = 75q2 = 0.0750.457150278.2255359C-D30.4470154801206153q3 = 35q3 = 0.0350.457150278.2255359D-G40.4470232660486057q4 = 25q4 = 0.0250.406160536.2707478G-J50.4470154801206153q5 = 90q5 = 0.0900.457120222.5804287J-I60.4470000072957691q6 = 150q6 = 0.1500.609300132.4056409I-H70.4470466694856617q7 = 175q7 = 01750.3041201708.9539397H-A80.4470466694856617q8 = 175q8 = 01750.3042803987.5591927B-E90.4470232660486057q9 = 50q9 = 0.0500.406160536.2707478C-F100.4470232660486057q10 = 40q10 = 0.0400.4061653.6270748E-F110.4470232660486057q11 = 15q11 = 0.0150.4061653.6270748F-G120.4470232660486057q12 = 15q12 = 0.0150.4061653.6270748E-I130.44703328769837464q13 = 25q13 = 0.0250.3551491.8100660Figure 2
Water supply network problem [3].


## 3.1.1. Modified Chun-Hui He’s Algorithm [24]

Step 1.
Necessary condition.(5)Fx0Fx1<0,where x0andx1 are the starting assumptions.Step 2.
Ancient Chinese algorithm computes(6)x2=x0−Fx0Rx0,x1,where Rx0,x1=Fx0−Fx1/x0−x1.Step 3.
Corrector step:(7)x3=x2−Fx2F′x2.Step 4.
Assumptions:(8)x0=x3−δ,x1=x3+δ,(9)δ=−Fx3F′x3.Step 5.
If we not get required accuracy, then go back to Step2.
Table1 provides the estimated fraction factor fn flow numbers using the modified Chun-Hui He’s Algorithm [24] for various pipe diameters (m).Table 1
The statistics values of flow, diameter, length, and constants value K displayed along with each value of the fraction factorf of each pipe in the network (Figure 2).
PipesNo.fnAssume flow (lt/s)Assume flow (m3/s)Diameter (m)Length (m)K = 8 fL/gD5п2A-B10.4470154801206153q1 = 125q1 = 0.1250.457120222.5804287B-C20.4470154801206153q2 = 75q2 = 0.0750.457150278.2255359C-D30.4470154801206153q3 = 35q3 = 0.0350.457150278.2255359D-G40.4470232660486057q4 = 25q4 = 0.0250.406160536.2707478G-J50.4470154801206153q5 = 90q5 = 0.0900.457120222.5804287J-I60.4470000072957691q6 = 150q6 = 0.1500.609300132.4056409I-H70.4470466694856617q7 = 175q7 = 01750.3041201708.9539397H-A80.4470466694856617q8 = 175q8 = 01750.3042803987.5591927B-E90.4470232660486057q9 = 50q9 = 0.0500.406160536.2707478C-F100.4470232660486057q10 = 40q10 = 0.0400.4061653.6270748E-F110.4470232660486057q11 = 15q11 = 0.0150.4061653.6270748F-G120.4470232660486057q12 = 15q12 = 0.0150.4061653.6270748E-I130.44703328769837464q13 = 25q13 = 0.0250.3551491.8100660Figure 2
Water supply network problem [3].


## 3.2. Loops Model in Water Network Topology

The system of nonlinear equations along each loop in this section was simulated.Loop I:WI,(10)Δw1+Δw9−Δw13−Δw7+Δw8=8gπ2f1l1q12d15+f9l9q92d95−f13l13q132d135−f7l7q72d75+f8l8q82d85,=222.5804287q12+536.2707478q92−91.8100660q132−1708.9539397q72+3987.5591927q82.Loop II:WII,(11)Δw2+Δw10−Δw11−Δw9=8gπ2f2l2q22d25+f10l10q102d105−f11l11q112d115−f9l9q92d95,=278.2255359q22+53.6270748q102−53.6270748q112−536.2707478q92.Loop III:WIII,(12)Δw3−Δw4+Δw12−Δw10=8gπ2f3l3q32d35−f4l4q42d45+f12l12q122d125−f10l10q102d105,=278.2255359q32−536.2707478q42+53.6270748q122−53.6270748q102.Loop IV:WIV,(13)−Δw5−Δw6+Δw13+Δw11−Δw12=8gπ2−f5l5q52d55−f6l6q62d65+f13l13q132d135+f11l11q112d115−f12l12q122d125.=−222.5804287q52−132.4056409q62+91.8100660q132+53.6270748q112−53.6270748q122.Above loop relations,WI,WII,WIII, andWIV can be written in matrix form:(14)100000−111000−101000000−11−100001−100000−10100000−1−100001−11×Δw1Δw2Δw3Δw4Δw5⋮Δw13=0,where l1,l2,…,l13 are the lengths of each pipe, d1,d2,…,d13 are the diameters, and water flow in each pipe, as shown in Figure 1, is q1,q2,…,q13, respectively. The positive sign showing flow direction is clockwise in left matrix of (14) and vice versa [5, 9, 30].


## 3.3. Nodes Model Topology

In this section, we will simulate the system of linear equation along each node by using Kirchhoff’s first law:(1)
Node A:−q1+q8−qA−output=0, where qA−output=50lt/s(2)
Node B:q1−q2−q9−qB−output=0, where qB−output=0(3)
Node C:q2−q3−q10−qC−output=0, where qC−output=0(4)
Node D:q3+q4−qD−output=0, where qD−output=60lt/s(5)
Node E:q9+q13−q11−qE−output=0, where qE−output=60lt/s(6)
Node F:q11+q10+q12−qF−output=0, where qF−output=70lt/s(7)
Node G:q5−q4−q12−qG−output=0, where qG−output=50lt/s(8)
Node H:−q7−q8+qH−input−qH−output=0, where qH−output=0,qH−input=350(9)
Node I:q7−q6−q13−qI−output=0, where qI−output=0(10)
Node J:q6−q5−qJ−output=0 where qJ−output=60lt/sand referent.Now, rewrite the above linear system of equations in matrix form [31]. The drawback of this matrix form is not a linear independent, that is why referent node would be omitted from the system of linear equation.(15)1000001000001−100000−1000001−100000−1000001100000000000000010−10100000000111000−11000000−1000000−1−1000000000−1100000−19×13×q1q2q3q4q5⋮q1313×1=qA−output=50qB−output=0qC−output=0qD−output=60qE−output=60qF−output=70qG−output=50qH−output−qH−input=−350qI−output=09×1.Above nodeJ count up as a referent, the left matrix in system (15) shows the rows corresponding to node A, node B, … node I, respectively.


## 3.4. Hardy Cross Method

The Hardy Cross method is applied to multiloop water network problems. Figures3 and 4 show the complete results of the HCM. This method converges to the required results after 24th iteration.Figure 3
The numerical results of head loss of equation (1) by using HCM and head loss in each loop going to zero after 24th iteration by selecting the random original flow in network problem.Figure 4
The numerical results of correction flow of equation (1) by using HCM and total corrected flow (m3/s) in each loop going to nil after 24th iteration with selecting random original flow in network problem, where ΔQI, ΔQII,ΔQIII, and ΔQIV change in flow of loop I–loop IV, respectively.The primary stage in resolution the problem is to make a net map display lengths and diameters and networks nodes. Later, we compose the original stream supply all through a pipe network. The selection of original flows must fulfill Kirchhoff’s law used in the system, and conservation law is also applicable for the entire network system [3, 13, 14, 27].Outcomes of HCM in column 7 indicate that after 24th iteration −Ww/Ww'(head loss) is nil by Darcy–Weisbach relation (1). The negative sign indicates anticlockwise water flows shown in Figure 1. Assign a sign with new flow of water. We will add in clockwise if ΔQ is +ve and vice versa. Similarly, we will subtract value if ΔQ is −ve and vice versa.


## 3.5. Newton Method

In this paragraph, hydrological problem is investigated by utilizing the NM having 2nd-order [32]. NM [23] or Newton-like procedure is not essential to take up an original estimate that fulfills the continuity law [33], as shown in Figure 1.Table2 provides the numerical results of correction flow of (1) by using conventional NM and change in two consecutive flows in each loop going to zero after 5th iteration by selecting the random original flow in network problem. The effectiveness of NM is better than HCM.Table 2
The water problem (Figure2) displaying the first iteration of technique NM ([32]). It is not necessary to follow Kirchhoff’s law in NM.
PipesLoopsDiameter (m)Length (m)qm3/sSign (q)Ww, equation (1)NM,qm3/sΔqm3/s10.4571200.12513.480.0892094−0.03579069I0.4061600.0511.340.0352368−0.0147632130.355120−0.025−1−0.490.1124880.08748870.304120−0.175−1−52.340.2107910.03579180.3042800.1751122.120.139209−0.035791LoopΣ74.110.036934220.4571500.07511.570.0539726−0.021027410II0.4061600.0410.860.017803−0.022197110.406150−0.015−1−0.110.08772490.072724990.406160−0.05−1−1.340.0352368−0.0147632LoopΣ0.970.014737330.4571500.03510.340.03616960.00116964III0.406160−0.025−1−0.340.0238304−0.0011696120.4061500.01510.11−0.0355278−0.0505278100.406160−0.04−1−0.860.017803−0.022197LoopΣ−0.74−0.072724850.457120−0.09−1−1.800.0383025−0.05169756IV0.609300−0.15−1−2.980.0983025−0.0516975130.3551200.02510.490.1124880.087488110.4061500.01510.110.08772490.0727249120.406150−0.015−1−0.11−0.0355278−0.0505278Σ−4.290.0062901All bold values are based on multidimension.


## 3.6. MNM (Modified Newton Method)

In this subsection, MNM achieves an optimum resolution of the nonlinear model simulated in Sections3.2 and 3.3 for scrutinizing the hydrological network considering. Present procedure is cost-effective and takes a smaller amount of time to attain the results than the HCM and conventional NM.Table3 provides the numerical results of correction flow of (1) by using MNM and change in two consecutive flows in each loop going to zero after three iterations by selecting the random original flow in network problem.Table 3
The first iteration of MNM ([34]) for the water problem (Figure 2).
PipesLoopsDiameter (m)Length (m)qm3/sSign (q)Ww, equation (1)MNM,qm3/sΔqm3/s10.4571200.12513.480.0879253−0.03707479I0.4061600.0511.340.0315287−0.0184713130.355120−0.025−1−0.490.1151210.09012170.304120−0.175−1−52.340.2120750.03707580.3042800.1751122.120.137925−0.037075LoopΣ74.110.03457520.4571500.07511.570.0563965−0.018603510II0.4061600.0410.860.0225996−0.0174004110.406150−0.015−1−0.110.086650.0716590.406160−0.05−1−1.340.0315287−0.0184713LoopΣ0.970.017174830.4571500.03510.340.0337969−0.00120314III0.406160−0.025−1−0.340.02620310.0012031120.4061500.01510.11−0.0392496−0.0542496100.406160−0.04−1−0.860.0225996−0.0174004LoopΣ−0.74−0.0716550.457120−0.09−1−1.800.0369535−0.05304656IV0.609300−0.15−1−2.980.0969535−0.0530465130.3551200.02510.490.115121−0.0134879110.4061500.01510.110.086650.07165120.406150−0.015−1−0.11−0.0392496−0.0542496Σ−4.29−0.1021805All bold values are based on multidimension.


## 4. Analysis and Results Discussion

Several processes must be carried out in order for this to operate. The recommended effort is run on a 10th Generation Intel Core i7 processor, 1 TB SSD, and GTX 1660 Ti (6 GB) graphics card with Windows 10 as the operating system. We used Mathematica 11.2 to do all of the existing simulations. The statistics in Table4 show only the first iteration of HCM, NM, and MNM, respectively. These results show that HCM reaches its ideal level after 24 rounds, NM after 5 iterations, and MNM after 3 iterations.Table 4
The first iteration of method HCM for the water problem (Figure2).
LoopsPipesDiameter (m)Length (m)qm3/sSign (q)Ww (equation (1))Ww′ (equation (2))CorrectionNewqm3/sI10.457120A1= 0.12513.4855.650.0904084390.406160A2= 0.0511.3453.63−0.0063207420.021729173130.355120A3=−0.025−1−0.4939.350.028735632−0.08832720170.304120A4=−0.175−1−52.34598.13−0.2095915780.304280A5= 0.1751122.121395.650.14040843LoopΣ74.112142.40−Ww/Ww'−0.0345915720.457150B1= 0.07511.5741.730.068679258II100.406160B2= 0.0410.8642.900.0070895850.026589673110.406150B3=−0.015−1−0.1115.080.028735632−0.05005637490.406160B4=−0.05−1−1.3453.63−0.03459157−0.021729173LoopΣ0.97153.35−Ww/Ww'−0.00632074230.457150C1= 0.03510.3419.480.042089585III40.406160C2=−0.025−1−0.3426.81−0.017910415120.406150C3= 0.01510.1115.080.028735632−0.006646047100.406160C4=−0.04−1−0.8642.90−0.006320742−0.026589673LoopΣ−0.74104.27−Ww/Ww'0.00708958550.457120−0.09−1−1.8040.06−0.061264368IV60.609300−0.15−1−2.9839.72−0.121264368130.3551200.02510.4939.35−0.034591570.088327201110.4061500.01510.1115.08−0.0063207420.050056374120.406150−0.015−1−0.1115.080.0070895850.006646047Σ−4.29149.30−Ww/Ww'0.028735632All bold values are based on multidimension.Figures5–8 show the evaluation and assessment of HCM, NM, and MNM in each loop, demonstrating that MNM is less expensive and takes less time to converge than traditional HCM and NM approaches. Table 5 provides the findings and evaluation of head loss in each loop of all approaches.Figure 5
The numerical compatibility of all HCM, NM, and MNM approaches in loop I.Figure 6
The numerical compatibility of all HCM, NM, and MNM approaches in loop II.Figure 7
The numerical compatibility of all HCM, NM, and MNM approaches in loop III.Figure 8
The numerical compatibility of all HCM, NM, and MNM approaches in loop IV.Table 5Start nodeEnd nodePipe indexHardy Cross method (m3/s)Newton method (m3/s)Modified NM (m3/s)AB10.08609120.08609120.0860912BC20.0579790.0579790.057979CD30.03808040.03808040.0380804DG4(a) −0.02191960.02191960.0219196GJ5−0.08546420.08546420.0854642JI6−0.1454640.1454640.145464IH7−0.2139090.2139090.213909HA80.1360910.1360910.136091BE90.02811220.02811220.0281122CF10−0.01989860.01989860.0198986EF110.0365569Hoboken0.0365569FG12−0.01354460.01354460.0135446EI130.06844470.06844470.0684447(a)The stream is moving in the opposite direction of the original flow, as indicated by the negative signs. The final water flow of HCM, NM, and MNM is explained in columns 7–9.The optimality status of HCM, NM, and MNM is given in Table5. We may also check that the original scheme, when compared to HCM and NM, achieves an optimal level with fewer steps and less time. On the other hand, HCM reaches the optimal stage after 24 rounds, NM after 5 iterations, and MNM after 3 iterations. The MNM’s key benefit is this statistic. This study’s novelty and effectiveness can be seen in these results.


## 5. Conclusion

The friction factorf is calculated using (3) in the first stage of this study. In a retraction domain Re > 4000, this equation represents the relationship between the tube’s innermost diameter, tube roughness, fraction factor, and Reynolds number. This relationship has an implied form that is impossible to solve explicitly. To circumvent this limitation, we numerically solved (3) using the most recent and up-to-date modified Chun-Hui He’s Algorithm [24] based on the sizes of each pipe in the system, as shown in Figure 2. The numerical value of the fraction factor was then used to calculate the water pressure function, which was then utilized to estimate the head loss and corrected flow between two consecutive values using the Hardy Cross method, Newton method, and modified Newton method. Tables 2–4 provide that HCM obtains the needed solution after 24 iterations, NM after 5 iterations, and MNM after 3 iterations, all with stopping criteria 1×10−10.



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*Source: 8116282-2022-02-15.xml*