Abstract
In this work, we applied the modified Chun-Hui He’s algorithm to evaluate for estimation of flow friction factor for value of friction factor by using Colebrook–White relation. The speedy, precise, and consistent evaluation of flow friction factor 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.
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. At , 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 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 Figure 1. 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.
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 have and and autonomous intersection points, i.e., 9 points and other one point is known as referent point and autonomous loops. In over problem, point is called referent node.
The Darcy–Weisbach equation and Colebrook–White relation for the Darcy friction factor can be used to analyze this pipe network [8, 28].where is the water pressure function. are the pipes length, diameter, and flow vectors, respectively, and is the gravity (m/s2).
Taking the first derivative of equation (1) and assumed as a variable herewith, we have
Darcy friction factor can be described as
Colebrook–White (3) is applicable only at turbulent flow in restricted domain at , where Re is called Reynolds number and is roughness height [25, 26, 28].
3.1. Modified Chun-Hui He’s Algorithm for Friction Factor
The quick, precise, and dependable evaluation of friction factor 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 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:
The friction factor f 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.where are the starting assumptions.
Step 2. Ancient Chinese algorithm computeswhere
Step 3. Corrector step:
Step 4. Assumptions:
Step 5. If we not get required accuracy, then go back to Step 2.
Table 1 provides the estimated fraction factor flow numbers using the modified Chun-Hui He’s Algorithm [24] for various pipe diameters (m).
3.2. Loops Model in Water Network Topology
The system of nonlinear equations along each loop in this section was simulated.
Loop I: ,
Loop II: ,
Loop III: ,
Loop IV: ,
Above loop relations, can be written in matrix form:where are the lengths of each pipe, are the diameters, and water flow in each pipe, as shown in Figure 1, is , 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: , where (2)Node B: , where (3)Node C: , where (4)Node D: , where (5)Node E: , where (6)Node F: , where (7)Node G: , where (8)Node H: , where (9)Node I: , where (10)Node J: where
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.
Above node J 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. Figures 3 and 4 show the complete results of the HCM. This method converges to the required results after 24th iteration.
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 (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 is +ve and vice versa. Similarly, we will subtract value if 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.
Table 2 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.
3.6. MNM (Modified Newton Method)
In this subsection, MNM achieves an optimum resolution of the nonlinear model simulated in Sections 3.2 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.
Table 3 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.
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 Table 4 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.
Figures 5–8 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.
The optimality status of HCM, NM, and MNM is given in Table 5. 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 factor 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
Nomenclature
| HCM: | Hardy Cross method |
| : | Ludolph’s number (=3.14159) |
| : | Friction function |
| : | Length (m/s) |
| : | Roughness height |
| : | Discharge pressure (l/s) |
| PVC: | Polyvinyl chloride |
| : | Diameter (m) |
| BVPs: | Boundary value problems |
| MNM: | Modified Newton method |
| : | 9.8 (m/s2) |
| : | Friction factor |
| NM: | Newton method |
| Re: | Reynolds number |
| : | Head loss |
| IT: | Iteration. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
W.A. and M.A. developed methodology and wrote and reviewed the article. W.A. investigated the study and wrote the original draft. W.A., A.J.R., and U.F. collected resources. Z.A. and F.F. collected data. M.A. conceptualized the study. W.A. and J.R. developed software. M.K. performed formal analysis.
Acknowledgments
The authors extend their appreciation to the deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work (IFP-2020-20).