Performance Evaluation of Selected Explicit Friction Models In Pipe Network Analysis

Performance Evaluation of Selected Explicit Friction Models In Pipe Network Analysis

Mmekutmfon Matthew Peter

Department of Chemical Engineering University of Uyo, Uyo, Nigeria

Sunday Boladale Alabi

Department of Chemical Engineering University of Uyo, Uyo, Nigeria

Abstract: This Accurate estimation of friction factor is essential for determining head loss and flow distribution in pipe networks. Although the Colebrook equation remains the benchmark, its implicit nature makes it computationally intensive. As a result, many explicit friction factor models have been developed as alternatives. This study evaluated the performance of 38 selected explicit models in solving pipe network problems using a non- linear gradient-based method in MATLAB. Four networks of increasing complexity (10, 24, 34, and 74 pipes) were used to assess each models computational efficiency, accuracy, numerical stability, and complexity. The Colebrook equation, solved using the Clamond method, served as the benchmark.

Across all networks, the number of iterations remained nearly constant for most models, but convergence times and computational efficiencies varied widely. Computational efficiencies ranged from 1.042 to 2.791 for network 1, 1.122 to

8.227 for network 2, 1.023 to 2.050 for network 3 and 1.001 to 1.794 for network 4. Although some explicit models, such as Swamee-Jain, Cojbasic Brkic A, Niazkar A, and Serghides A, showed good performance, the Colebrook equation remained the fastest and most stable across all networks.

Error analysis using mean square error (MSE) across four pipe networks showed that Niazkar A, Cojbasic Brkic A, and Serghides A consistently produced the lowest errors and they closely matched Colebrooks results in both nodal heads and flow rates. For these models, nodal head MSE values ranged from

6.26×10¹² to 6.75×10¹² and flow rate MSE values from 6.61×1011 to 6.17×1010 in Network 1; 1.415×10¹ to 1.3529×1016 for nodal head and 3.980×1019 to 9.579×1018 for flow rate in Network 2;

1.14×1010 to 1.88×105 for nodal head and 2.179×1014 to 1.0795×1012 for flow rate in Network 3; and 2.46×1018 to

5.43×1018 for nodal head and 2.851×1018 to 4.449×1018 for flow rate in Network 4. In contrast, models such as Avci Karagoz, Buzzelli, and Fang exhibited convergence failures in the most complex network, indicating numerical instability in highly interconnected systems.

In conclusion, while a few explicit models are suitable alternatives in specific scenarios, the Colebrook equation remains the most reliable choice for pipe network analysis.

Keywords: Pipe Network Analysis, Friction Factor, Colebrook Equation, Explicit Models, Hydraulic Engineering

1. INTRODUCTION

   A pipe network is a system of interconnected pipes designed for transporting fluids such as water, oil, and gas. Pipe networks are vital to modern infrastructure, including water distribution, oil and gas transport, and HVAC systems

   (Chaudhry, 2014). Their efficiency depends on accurately predicting fluid behaviour, particularly frictional losses that occur due to interactions between the fluid and pipe walls, which cause energy dissipation and pressure reduction along the pipeline. Correct estimation of these losses ensures optimal system performance and resource utilization (García, 2022).

   In pipe systems, head loss in turbulent flow is commonly determined using the DarcyWeisbach relation given in Equation (1):

   h = × L/D × V2/2g (1)

   where h is head loss, L is pipe length, D is diameter, V is velocity, g is gravitational acceleration, and is the friction factor.

   The friction factor (f), which accounts for resistance due to pipe roughness, flow velocity, and fluid properties, is most accurately defined by the ColebrookWhite (CW) model (Equation 2):

   1/ = -2log10 ((/D)/3.7 + 2.51/(Re)) (2)

   Although accurate, the ColebrookWhite or simply Colebrook equation is implicit in f, meaning it cannot be solved directly and requires iterative computations. This implicit nature of f adds complexity to hydraulic analyses and underscores the importance of accurately estimating its value for precise predictions of frictional losses in pipe networks. To overcome this limitation, several explicit friction factor relations have been proposed to approximate the Colebrook White model, each with varying degrees of performance across different criteria such as computational efficiency, accuracy, numerical stability and model complexity.

   Although numerous explicit friction factor relations exist, the majority of their evaluations in past studies have been carried out without applying them to pipe networks, which are the real environment for head-loss analysis. Very limited works have been reported on their evaluation when used specifically in pipe network analysis. Over the years, researchers have compared these explicit models under various conditions. Niazkar and Talebbeydokhti (2019) applied non-linear solution methods in the evaluation of explicit friction factor relations on pipe networks, where number of iterations was used as a measure of computational speed of these models. Using number of iterations as a measure of computational speed among other factors, may be erroneous as some relations may take more time to converge but with smaller or the same number of iterations.

   Udoh (2023) attempted to evaluate the performance of several explicit friction factor relations, primarily focusing on simple pipe networks and comparing their performances with that of Colebrook equation. He found that Colebrook equation converges faster than most explicit models when applied to simple pipe networks using linear solution method. This finding suggested that there may not be the need for explicit models in such scenarios, as the Colebrook equation demonstrated superior convergence performance. However, since Udohs analysis was limited to simple networks and linear solution method, it is not clear what the performances of these explicit models would be, in more complex systems or when non-linear solution methods are employed.

   Consequently, in this study, based on a number of criteria such as accuracy, convergence speed, numerical stability, and computational efficiency, a comprehensive performance evaluation of selected explicit friction factor equations when applied to pipe networks of varying complexity using a non- linear gradient-based solution method was done. The remaining sections present the methodology, followed by the results and discussion, and finally the conclusions.

2. COMPUTATIONAL METHODOLOGY
   1. Pipe Networks

      Four different pipe networks of increasing complexity were obtained from the literature and were analyzed to assess the performance of 38 selected explicit friction factor relations. The first pipe network consists of 10 pipes and 7 nodes (Figure 1). The second pipe network consists of twenty-four pipes and seven nodes (Figure 2). The third pipe network consists of 34 pipes and 32 nodes, and is fed by gravity from a reservoir with a 100 m fixed head (Figure 3). The fourth pipe network contains 74 pipes and 48 nodes (Figure 4).

      Fig. 1. Network 1 (Source: Jeppson (1974))

      Fig. 2. Network 2 (Source: Ciaponi et al. (2015))

      Fig. 3. Network 3 (Source: Fujiwara and Khang (1990))

      Fig. 4. Network 4 (Source: Chin et al. (1978))

   2. Friction Factor Relations

      In this study, 38 explicit friction factor models were evaluated to compare their performnce against the Colebrook White (CW) equation. These 38 models were chosen to provide a representative mix of the most established, widely referenced, and recently developed explicit friction factor equations, enabling a fair and comprehensive comparison. They include relations developed by Avci and Karagoz (2009), Azizi (2008), Barr (1981), Beluco and Schettini (2001), Biberg (2017), Brki (2011), Brki and Parks (2019a), Brki and Parks (2011b), Brki and Parks (2011c), Buzzelli (2008), Chen (1979), Churchill (1977), Cojbasic and Brki (2013a), Cojbasic and Brki (2013b), Eck (1973), Fang et al. (2011), Ghanbari et al. (2011), Haaland (1983), Jain (1976), Swamee and Jain (1976), Li et al. (2011), Manadili (1997), Niazkar (2019a), Niazkar (2020b), Offor and Alabi (2016), Papaevangelou et al. (2010), Rao and Kumar (2010), Romeo et al. (2002), Round (1980), Serghides (1984a), Serghides (1984b), Shacham et al. (1980), Shaikh (2012), Sonnad and Goudar (2006), Vatankhah (2018), Vatankhah and Kouchakzadeh (2008), Zigrang and Sylvester (1982a) and Zigrang and Sylvester (1982b).

      The ColebrookWhite equation was used as the benchmark for all comparative analyses, and the solution was computed using the Clamond (2009) method, which provides a fast and accurate iterative solution for the implicit relation.

   3. Evaluation Criteria

      The performance of each explicit model was assessed using four key criteria. These comparison indices help highlight the strengths and limitations of each model, particularly when applied in computational frameworks for fluid flow analysis.

      The most commonly used criteria in the literature are outlined below:

      1. Accuracy: This was evaluated by comparing each models nodal head and pipe flow results against those derived from the Colebrook equation. To quantify this closeness, Mean Square Error (MSE) criterion was used. MSE calculates the average of the squares of the differences between the models predicted values and those from the Colebrook equation. It penalizes larger errors more than smaller ones, making it useful for identifying models that deviate significantly under certain conditions. A lower MSE indicates a better overall fit to the reference values.
      2. Computational Efficiency: Computational efficiency was calculated as the ratio of the model’s convergence time (seconds) to that of the Colebrook equation. A value greater than one (1) indicates that the model is slower than the Colebrook equation while the value less than one (1) indicates that the model is faster than the Colebrook equation and a value equal to one (1) means the model is exactly as fast as the Colebrook equation.
      3. Numerical Stability: Stability was judged based on whether the solver successfully converged for each network. Models that failed to converge, particularly in larger or more complex systems, were classified as numerically unstable.
      4. Number of Iterations to Convergence: The total number of iterations required by the solver to converge was recorded for each model. This helped in identifying whether faster convergence corresponds to reduced computational cost.
   4. Computational Procedure

   The h-based gradient method of solution was used in analyzing the four complex pipe networks. This method applies the Newton-Raphson technique in terms of pipe flows and nodal heads to obtain a simultaneous solution to the mass and energy balance system of equations. The pipe networks were solved through an iterative solution of a system of non-linear equations. The pipe properties, fluid properties and other data needed to start the analysis were inputted into Excel spreadsheet. The gradient algorithm method was coded into MATLAB using appropriate formulated codes. The formulated code was designed to call in the input data from the Excel spreadsheet into MATLAB environment.

   The results of the analysis were displayed and the best- performing relations were selected based on number of iterations, computational time, accuracy, and stability of the iteration scheme. The displayed results include head losses, flow rates, number of iterations, and computational time taken by each relation. The obtained results from each explicit friction factor relation were compared with the results from the Colebrook solution. The error for each explicit relation was computed for both nodal heads and flow rates using mean square error as an accuracy metric. The error measure was used to comprehensively evaluate the performance of each explicit friction factor relation.

3. RESULTS AND DISCUSSION
   1. Solver Convergence and Computational Performance

      This section presents the results of the iteration count, computational time, and computational efficiency for each of the four pipe networks. These three metrics are combined into a single table per network, allowing a clearer comparison of solver performance for each explicit friction factor relation.

      1. Network 1 (10 Pipes): The computational performances of the explicit models for Network 1 are as shown in Table I. It is obvious that all the models converged successfully with nearly the same number of iterations. However, the total computational time vary due to differences in the model complexity.

         Models such as Biberg, Swamee Jain, Vantankhah A, Serghides A and Niazkar A demonstrated the fastest convergence among the explicit relations as shown in Table 1, while Colebrook remained the overall fastest. The computational efficiency results show that all the explicit relations had efficiency ratios slightly greater than 1, indicating that they were slower than the Colebrook relation, although the differences were relatively small for the top-performing models. This overall result suggests that the number of iterations alone is not a reliable indicator of a models overall computational performance in pipe network analysis. This finding directly contradicts the conclusion of Niazkar and Talebbeydokhti (2019), who suggested that iteration count reflects performance (convergence speed).

         TABLE I. Convergence and Computational Performance of

         5

         Friction Model	Number of Iteration	Computational Time (s)	Computational Efficiency
         Avci Karagoz	5	0.0279799	1.1704279
         Azizi	5	0.0421797	1.7644202
         Barr	5	0.0296128	1.2387338
         Beluco Schettini	6	0.0667423	2.7918990
         Biberg	5	0.0249097	1.0419983
         Brkic	5	0.0373603	1.5628197
         Brkic Parks A	5	0.0295799	1.2373576
         Brkic Parks B	5	0.0373603	1.5628197
         Brkic Parks C	5	0.0319264	1.3355141
         Buzzelli	5	0.0296187	1.2389806
         Chen	5	0.0363211	1.5193489
         Churchill	5	0.0525646	2.1988312
         Cojbasic Brkic A	5	0.0356075	1.4894983
         Cojbasic Brkic B	5	0.0373603	1.5628197
         Eck	7	0.0641269	2.6824941
         Fang	0.0298192	1.2473677
         Ghanbari	5	0.0314821	1.3169285
         Haaland	5	0.0538186	2.2512873
         Jain	5	0.0356453	1.4910795

          

         Network 1 (10 pipes)

         Friction Model	Number of Iteration	Computational Time (s)	Computational Efficiency
         Li	6	0.0604117	2.5270834
         Manadili	5	0.0356075	1.4894983
         Niazkar A	5	0.0274535	1.1484081
         Niazkar B	5	0.0362512	1.5164249
         Offor Alabi	5	0.0303333	1.2688731
         Papaevangelou	5	0.0330365	1.3819507
         Rao Kumar	5	0.0383431	1.6039312
         Romeo	5	0.0295799	1.2373576
         Round	5	0.0275689	1.1532354
         Serghides A	5	0.0267423	1.1186578
         Serghides B	6	0.0653564	2.7339253
         Shacham	5	0.0337999	1.4138845
         Shaikh	5	0.0392573	1.6421732
         Sonnad Goudar	5	0.0442266	1.8500441
         Swamee Jain	5	0.0258499	1.0813278
         Vantankhah A	5	0.0263246	1.1011850
         Vantankhah- Kouchakzadeh	5	0.0282276	1.1807895
         Zigrang-Sylvester	5	0.0547462	2.2900898
         Zigrang-Sylvester B	5	0.0543566	2.2737924
         Colebrook	5	0.0239057

      2. Network 2 (24 Pipes): The computational performances of the explicit models for Network 2 are as shown in Table II. It is obvious that all the explicit models converged within similar iteration ranges (19 to 34), but the computational times increased compared to Network 1, which is expected because the larger network contains more pipes and nodes, resulting in more head-loss evaluations and matrix updates per iteration. Models such as Brkic Parks A, Cojbasic Brkic-A, Romeo, and Niazkar A, recorded shorter convergence times compared to other explicit relations, though still slower than the Colebrook equation. The computational efficiencies for these models were greater than 1, showing that they were slower than the Colebrook reference. This study confirms that, for this network, iteration count does not correlate with computational time, and therefore cannot be used as a reliable indicator of solver performance.

         A consistent trend observed in this network is the influence of model complexity, measured by the number of internal iterations. Models with higher internal complexity, such as Serghides A (10 internal iterations), Niazkar A (6 internal iterations), Biberg (4 internal iterations), and Vantankhah A (4 internal iterations), achieved shorter computational times and therefore exhibited better computational efficiency. In contrast, low-complexity models such as Azizi (1 internal iteration), BelucoSchettini (1 internal iteration), Eck (1 internal iteration), etc. despite having the simplest algebraic structures, recorded higher convergence times. This reinforces the finding that model simplicity does not translate to computational speed within network solvers.

         TABLE II. Convergence and Computational Performance of

         Network 2 (24 pipes)

         Friction Model	Number of Iteration	Computational Time (s)	Computational Efficiency
         Avci Karagoz	19	0.3549771	1.3603506
         Azizi	30	0.7049535	2.7015374
         Barr	19	0.4102749	1.5722639
         Beluco Schettini	31	0.4215987	1.6156592
         Biberg	19	0.314825	1.2064789
         Brkic	19	0.4230497	1.6212198
         Brkic Parks A	19	0.2928519	1.1222731
         Brkic Parks B	19	0.4793162	1.8368455
         Brkic Parks C	19	0.3509846	1.3450504
         Buzzelli	20	0.446381	1.7106305
         Chen	19	0.3021765	1.1580070
         Churchill	19	0.3758147	1.4402049
         Cojbasic Brkic A	19	0.2989579	1.1456726
         Cojbasic Brkic B	33	1.106872	4.2417778
         Eck	19	0.3053319	1.1700992
         Fang	35	1.3462864	5.1592667
         Ghanbari	19	0.3994126	1.5306372
         Haaland	31	0.4526304	1.7345796
         Jain	19	0.4361855	1.6715591
         Li	19	0.3004665	1.1514539
         Manadili	35	2.1467053	8.22664865
         Niazkar A	19	0.3001856	1.1503774
         Niazkar B	19	0.4052987	1.5531940
         Offor Alabi	19	0.4110778	1.5753408
         Papaevangelou	19	0.4075448	1.5618016
         Rao Kumar	19	0.4212987	1.6145096
         Romeo	19	0.2990844	1.1461574
         Round	33	0.4545118	1.7417895
         Serghides A	19	0.3089729	1.1840523
         Serghides B	19	0.3075095	1.1784442
         Shacham	19	0.360168	1.3802432
         Shaikh	29	0.3477332	1.3325903
         Sonnad Goudar	19	0.3361595	1.2882374
         Swamee Jain	19	0.3108572	1.1912734
         Vantankhah A	19	0.3974598	<>1.5231537
         Vantankhah- Kouchakzadeh	19	0.3689207	1.4137855
         Zigrang-Sylvester	19	0.4043953	1.5497320
         Zigrang-Sylvester B	19	0.3906788	1.4971674
         Colebrook	19	0.2609453

         Friction Model	Number of Iteration	Computational Time (s)	Computational Efficiency
         Haaland	20	0.2370852	1.0604598
         Jain	20	0.2684554	1.2007757
         Li	20	0.2476071	1.1075232
         Manadili	20	0.3653969	1.6343860
         Niazkar A	20	0.2324601	1.0397721
         Niazkar B	20	0.2604652	1.1650363
         Offor Alabi	20	0.2844972	1.2725292
         Papaevangelou	20	0.2830856	1.2662152
         Rao Kumar	20	0.2849943	1.2747527
         Romeo	20	0.2530465	1.1318532
         Round	23	0.2898574	1.2965049
         Serghides A	20	0.2287812	1.0233168
         Serghides B	20	0.2462684	1.1015354
         Shacham	20	0.3731537	1.6690814
         Shaikh	20	0.2897913	1.2962092
         Sonnad Goudar	20	0.3170146	1.4179765
         Swamee Jain	20	0.2305696	1.0313161
         Vantankhah A	20	0.2477378	1.1081079
         Vantankhah- Kouchakzadeh	20	0.2468879	1.1043063
         Zigrang-Sylvester	20	0.2573534	1.1511175
         Zigrang-Sylvester B	20	0.2670543	1.1945087
         Colebrook	20	0.2235683

          

      3. Network 3 (34 Pipes): The computational performances of the explicit models for Network 3 are as shown in Table III. This result shows that although the explicit models converged within a relatively narrow iteration band (mostly 2024 iterations), the convergence times varied substantially, ranging from 0.2288 s to 0.4584 s. This confirms again that iteration count does not correlate with computational time, as several models with identical iteration counts produced widely different convergence times.

         Models such as Serghides A, Biberg, Swamee Jain, Niazkar A and Cojbasic Brkic A were among the top 5 performers, converging more rapidly and efficiently than most models. A closer inspection shows that there is no simple, monotonic relationship between the number of internal iterations and convergence time. High-complexity models (in terms of number of internal iterations), such as Serghides A, Cojbasic Brkic A, Niazkar A, achieved fast convergence, but relations such as SwameeJain, and Biberg, which have lower number of internal iterations also recorded short runtimes.

         In contrast, some of the simpler, low-complexity models such as Azizi, AvciKaragoz, BelucoSchettini, and Chen exhibited much longer convergence times. This confirms that simpler expressions do not necessarily compute faster in network simulations; the internal numerical stability and structure matter more than algebraic simplicity.

         Despite the strong performance of several explicit relations, the Colebrook equation again showed the fastest convergence for Network 3 (0.2236 s), confirming that even with increasing network complexity, the implicit formulation remains computationally superior when solved using the Clamond method.

         TABLE III. Convergence and Computational Performance of

         Network 3 (34 pipes)

         Friction Model	Number of Iteration	Computational Time (s)	Computational Efficiency
         Avci Karagoz	24	0.4584391	2.0505550
         Azizi	23	0.3500506	1.5657434
         Barr	20	0.3094163	1.3839900
         Beluco Schettini	23	0.3592821	1.6070350
         Biberg	20	0.2288322	1.0235449
         Brkic	20	0.4007588	1.7925564
         Brkic Parks A	20	0.2484558	1.1113194
         Brkic Parks B	20	0.2815977	1.2595600
         Brkic Parks C	20	0.2404098	1.0753304
         Buzzelli	20	0.3134542	1.4020511
         Chen	23	0.4105554	1.8363757
         Churchill	20	0.3691923	1.6513624
         Cojbasic Brkic A	20	0.2358404	1.0548919
         Cojbasic Brkic B	20	0.3634816	1.6258190
         Eck	20	0.3214455	1.4377955
         Fang	20	0.2577506	1.1528942
         Ghanbari	20	0.3607775	1.6137238

      4. Network 4 (74 Pipes): This network represented the most complex case analysed in this work. As shown in Table 4, almost all explicit models converged within 41 to 45 iterations, indicating that the iteration count remained consistent despite the larger system size. However, three models such as Avci Karagoz, Buzzelli, and Fang, failed to converge, demonstrating numerical instability when applied to a highly interconnected network.

         Although the iteration counts were nearly identical, the computational times varied significantly, ranging from approximately 0.339s to 0.608s among models that converged. This reinforces the established finding that iteration count does not correlate with computational speed, especially in complex networks. Among the convergent models, Serghides A, Cojbasic Brkic A, Brkic Parks C and Niazkar A continued to show superior performance, though still slower than the Colebrook equation.

         A closer look at model complexity reveals that there is no direct relationship between the number of internal iterations and computational speed. High-complexity models (in terms of number of internal iterations), such as Serghides A and Cojbasic Brkic A, delivered the fastest runtimes, while low- complexity models such as Azizi, Beluco Schettini, Eck, were among the slowest. This shows that in a gradient-based solver, numerical behaviour and structural stability of a model determines practical perfomance.

         TABLE IV: Convergence and Computational Performance of

         Friction Model	Number of Iteration	Computational Time (s)	Computational Efficiency
         Avci Karagoz	–	–	–
         Azizi	41	0.4894358	1.4440888
         Barr	41	0.3929686	1.1594607
         Beluco Schettini	45	0.6083332	1.7948977
         Biberg	41	0.3770132	1.1123840
         Brkic	41	0.502949	1.4839598
         Brkic Parks A	41	0.4816927	1.4212427
         Brkic Parks B	41	0.444987	1.3129419
         Brkic Parks C	41	0.3509718	1.0355484
         Buzzelli	–	–	–
         Chen	41	0.469168	1.3842883
         Churchill	41	0.3833203	1.1309932
         Cojbasic Brkic A	41	0.3401325	1.0035668
         Cojbasic Brkic B	42	0.5678299	1.6753920
         Eck	41	0.54776	1.6161754
         Fang	–	–	–
         Ghanbari	41	0.3704834	1.0931177
         Haaland	41	0.4175989	1.2321328
         Jain	42	0.5090967	1.502099
         Li	42	0.5043876	1.488204
         Manadili	41	0.4364567	1.287773
         Niazkar A	42	0.3617754	1.067425
         Niazkar B	41	0.4533966	1.337755
         Offor Alabi	41	0.5151997	1.5201057
         Papaevangelou	41	0.4960696	1.463662
         Rao Kumar	41	0.4254019	1.2551557
         Romeo	41	0.5349866	1.5784873
         Round	41	0.3962113	1.1690283
         Serghides A	41	0.3393895	1.0013745
         Serghides B	42	0.5452413	1.6087439
         Shacham	41	0.4435457	1.3086893
         Shaikh	41	0.3821235	1.1274621
         Sonnad Goudar	41	0.4115435	1.2142662
         Swamee Jain	41	0.3749543	1.1063092
         Vantankhah A	41	0.4522207	1.3342850
         Vantankhah- Kouchakzadeh	41	0.4768442	1.4069371
         Zigrang-Sylvester	41	0.3782463	1.1160223
         Zigrang-Sylvester B	41	0.3864233	1.1401486
         Colebrook	41	0.3389236

          

         Network 4 (74 pipes)

         Summary, this study shows that the number of iterations is not a valid indicator of computational efficiency. All the explicit friction factor models tested converged with nearly the same number of iterations for each of the four networks, yet they produced varying convergence times due to differences in mathematical complexity. This study also shows the Clamond time function which was used to evaluate the Colebrook equation converged faster than all the explicit models across the four networks. Consequently, there may not be any need for the use of explicit models except there is a problem of numerical instability from the use of the Colebrook equation. In this study, however, the Colebrook equation, solved using the Clamond method did not exhibit any numerical instability across all four network case studies.

   2. Accuracy

      The nodal heads and flowrates errors obtained using each explicit model were compared to those derived from the Colebrook equation, which serves as the benchmark. Mean square error metric was used as a measure of accuracy. The performance of each model under mean square error metrics was evaluated across all four pipe networks.

      1. Mean Square Nodal Error: Table V shows the mean square nodal error for each model across the four pipe networks. Across all four pipe networks, the mean square nodal error results revealed that only a few explicit friction factor models consistently achieved high accuracy when compared to the Colebrook equation. Models with lower errors demonstrated stable and precise behavior even in larger or more complex networks, while others, especially those with higher errors exhibited reduced reliability as network complexity increases.
         

         TABLE V: Mean Square Nodal Error across All the Four Networks

         Friction Model	Network 1	Network 2	Network 3	Network 4
         Avci Karagoz	1.79E-10	260.84	22789	–
         Azizi	7.50E-11	2.424	18262.7	10063.01
         Barr	1.16E-11	0.009	5881.7	1.3442
         Beluco Schettini	8.96E-12	2.403	18186.2	9968.61
         Biberg	7.58E-12	5.35E-07	0.012	4.46E-05
         Brkic	1.24E-08	7.68E-06	15.294	0.0245
         Brkic Parks A	1.02E-10	7.69E-08	0.0106	0.0070
         Brkic Parks B	1.74E-11	2.12E-07	0.0407	0.0084
         Brkic Parks C	6.87E-11	1.04E-06	5.42E-05	0.0091
         Buzzelli	1.17E-11	0.062	1176.05	–
         Chen	1.25E-11	1.16E-05	0.0253	0.0006
         Churchill	1.31E-11	0.0002	0.4194	0.1664
         Cojbasic Brkic A	6.75E-12	8.94E-17	1.35E-05	2.46E-18
         Cojbasic Brkic B	1.74E-11	32.813	248851.6	136037.8
         Eck	6.06E-12	0.0003	123.3	0.0572
         Fang	4.26E-11	19.467	8237.3	–

         Friction Model	Network 1	Network 2	Network 3	Network 4
         Ghanbari	2.06E-11	0.0004	3.9327	0.4364
         Haaland	1.26E-11	2.406	18277.3	9967.7
         Jain	1.50E-11	0.0001	0.7983	0.0672
         Li	4.29E-11	0.0018	548.112	0.2044
         Manadili	9.51E-12	1.1014	303.73	63.4013
         Niazkar A	6.31E-12	1.41E-17	1.14E-10	5.43E-18
         Niazkar B	3.60E-11	0.0007	17.89	0.2858
         Offor Alabi	1.14E-11	0.0002	0.0148	0.1759
         Papaevangelo u	1.25E-11	7.69E-05	0.0707	0.0191
         Rao Kumar	4.54E-11	0.0086	5884.5	1.4240
         Romeo	6.87E-11	4.16E-08	0.0213	0.0027
         Round	4.59E-11	2.5385	18383.4	10738.7
         Serghides A	6.26E-12	1.35E-16	1.88E-05	3.67E-18
         Serghides B	6.39E-12	5.04E-10	0.23786	5.68E-10
         Shacham	5.58E-12	5.45E-08	1.5340	0.0033
         Shaikh	5.37E-11	2.156	11257.2	9748.22
         Sonnad Goudar	9.38E-12	2.32E-06	0.37839	0.00023
         Swamee Jain	7.26E-12	0.0002	0.5979	0.1616
         Vantankhah A	2.61E-12	1.30E-07	0.00150	0.1616
         Vantankhah- Kouchakzade h	1.15E-12	1.07E-08	0.30172	6.52E-06
         Zigrang- Sylvester	1.83E-11	1.70E-10	0.02394	2E-10
         Zigrang- Sylvester B	3.26E-11	9.46E-06	24.6564	0.0001

         In Network 1, which consists of 10 pipes, the best- performing models were Vantankhah-Kouchakzadeh, Vantankhah A, Eck, and Serghides A. Models such as Niazkar A, Serghides B and Cojbasic Brkic A relations were the next best explicit relations, in no particular order. In Network 2, which consists of 24 pipes, the best-performing models were Niazkar A, Cojbasic Brkic A, and Serghides A. In Network 3, which consists of 34 pipes, the best-performing models were were Niazkar A, Cojbasic Brkic A, and Serghides A. In Network 4, which consists of 74 pipes, the best-performing models were Cojbasic Brkic A, Serghides A and Niazkar A.

         Models such as Niazkar A, Cojbasic Brkic A, and Serghides A consistently exhibited the lowest MSE values across varying network sizes, indicating their ability to maintain high accuracy in predicting nodal heads. For instance, in smaller networks, these models produced mean square errors close to zero, and even in larger systems, they maintained minimal deviation from the Colebrook benchmark.

         In contrast, models like Avci Karagoz, Azizi, Round, Cojbasic Brkic B, performed poorly as their mean square errors increase with the network size, confirming their limited applicability in complex or highly interconnected systems. Overall, the best- performing models with respect to accuracy across the four networks was Niazkar A and Cojbasic Brkic A.

      2. Mean Square Flow Error: Table VI shows the mean square flow rate error for each model across the four pipe networks. Across all the four networks, the Mean Square Error values reveal notable variations in the accuracy of the explicit friction factor models. In general, models such as Niazkar A, Cojbasic Brkic A, and Serghides A, consistently record the lowest MSE values, indicating superior performance in estimating flowrate relative to the Colebrook equation.
         

         Friction Model	Network 1	Network 2	Network 3	Network 4
         Avci Karagoz	9.173E-09	11.225	9.86E-07	–
         Azizi	8.664E-09	0.0259	3.92E-05	1.265E-05
         Barr	1.471E-08	0.00011	0.0013	0.1085
         Beluco Schettini	6.475E-09	0.02552	2.3E-05	4.108E-05
         Biberg	1.234E-11	6.613E-09	2.24E-11	6.279E-06
         Brkic	8.627E-09	2.475E-07	1.71E-06	0.00044
         Brkic Parks A	4.787E-09	2.315E-09	5.09E-11	3.747E-05
         Brkic Parks B	8.627E-09	4.914E-09	2.61E-08	2.715E-05
         Brkic Parks C	4.298E-09	1.355E-08	2.58E-10	1.392E-05
         Buzzelli	1.471E-08	0.0012	0.0003	–
         Chen	1.651E-08	1.39E-07	8.30E-08	0.00039
         Churchill	1.302E-08	2.054E-06	3.57E-06	0.00285
         Cojbasic Brkic A	6.17E-10	6.736E-18	7.21E-14	2.851E-18
         Cojbasic Brkic B	8.627E-09	0.3835	1.44E-11	1.758E-05
         Eck	4.235E-10	5.662E-06	0.0001	0.0048
         Fang	2.018E-09	0.24416	0.0083	–
         Ghanbari	1.782E-07	4.281E-06	4.88E-06	0.01181
         Haaland	1.039E-08	0.02555	6.283E-06	0.00199
         Jain	1.186E-08	1.422E-06	4.09E-06	0.0032
         Li	2.233E-09	2.316E-05	0.0001	0.02371
         Manadili	6.168E-09	0.00652	0.0006	6.1628
         Niazkar A	6.61E-11	3.9801E-19	2.18E-14	4.449E-18
         Niazkar B	3.912E-09	8.471E-06	2.59E-06	0.10278
         Offor Alabi	1.834E-08	2.927E-06	5.76E-08	0.00358
         Papaevangelo u	8.252E-09	8.898E-07	9.22E-08	0.00099
         Rao Kumar	1.361E-08	0.00011	0.0013	0.10767
         Romeo	4.298E-09	1.1567E-09	4.89E-09	1.577E-05
         Round	3.415E-05	0.0276	1.50E-05	0.8689
         Serghides A	6.98E-11	9.580E-18	1.08E-12	4.152E-18
         Serghides B	1.249E-08	2.083E-11	2.07E-07	4.575E-10
         Shacham	1.757E-08	3.36E-09	1.53E-07	1.057E-05
         Shaikh	2.517E-08	0.023248	0.0029	2.90004
         Sonnad Goudar	3.509E-09	3.035E-08	4.30E-07	1.759E-05

          

         TABLE VI: Mean Square FLOW Error across All the Four Networks

         Friction Model	Network 1	Network 2	Network 3	Network 4
         Swamee Jain	1.097E-08	1.949E-06	3.83E-06	0.00270
         Vantankhah A	1.322E-08	4.09E-09	4.97E-10	0.00270
         Vantankhah- Kouchakzade h	1.830E-08	1.959E-10	3.28E-07	1.904E-06
         Zigrang- Sylvester	1.104E-08	5.063E-12	6.89E-09	1.368E-10
         Zigrang- Sylvester B	6.312E-09	1.522E-07	2.54E-05	3.458E-05

         In Network 1, which consists of 10 pipes, the best- performing models were Biberg, Niazkar A, Serghides A. Eck, Cojbasic Brkic A, Fang and Li relations were the next best explicit relations, in no particular order showing in Table 4. In Network 2, which consists of 24 pipes, the best-performing models were Niazkar A, Cojbasic Brkic A, and Serghides A. In Network 3, which consists of 34 pipes, the best-performing models were were Niazkar A, Cojbasic Brkic A, and Serghides

         1. In Network 4, which consists of 74 pipes, the best- performing models Cojbasic Brkic A, Serghides A and Niazkar A.

   Models such as Niazkar A, Cojbasic Brkic A, and Serghides A consistently exhibited the lowest MSE values across varying network sizes, indicating their ability to maintain high accuracy in predicting flow heads. For instance, in smaller networks, these models produced mean square errors close to zero, and even in larger systems, they maintained minimal deviation from the Colebrook benchmark.

   In contrast, models like Avci Karagoz, Azizi, Cojbasic Brkic B which performed poorly in mean square flow error evaluations, also showed relatively high MSE values, confirming their limited applicability in complex or highly interconnected systems. Overall, the best-performing models with respect to mean square flow error across the four networks was Niazkar A and Cojbasic Brkic A.

   Overall, when both convergence behaviour and error metrics are considered together, models such as Niazkar A, CojbasicBrkic A, and Serghides A consistently offered the best balance of speed and accuracy. These models recorded some of the best computational efficiency values while simultaneously producing the smallest nodal head and mean square flow errors, and their performance remained stable as network size increases. Models such as AvciKaragoz, Cojbasic Brkic B show comparatively high computational times and consistently larger flow-error magnitudes, indicating weaker suitability for network-based hydraulic analysis. Although few explicit models demonstrated good performance, the ColebrookWhite equation remained superior, consistently exhibiting faster convergence, consistent numerical stability, and higher accuracy.

4. CONCLUSION

This study evaluated the performance of 38 explicit friction factor equations within iterative pipe network solvers using MATLAB across four networks of increasing complexity. The results revealed notable differences in accuracy, numerical stability, and computational efficiency among the models when applied during pipe networks analysis.

A key finding is that the number of iterations required for convergence is not a reliable measure of computational efficiency. Although most models converged in a similar number of iterations, their computation times varied significantly due to differences in their mathematical complexity, particularly the presence of multiple logarithmic or exponential terms.

Among the selected models, Niazkar A, Cojbasic Brkic A, and Serghides A consistently delivered accurate and stable results across all networks. Their ability to balance computational speed and reliability makes them suitable for practical engineering applications.

In contrast, models like AvciKaragoz, Buzzelli, and Fang failed to converge in the most complex network, suggesting that their structure might have made them prone to numerical instability when applied to large pipe networks.

Despite the good performance of a few explicit models, the Colebrook-White equation remained superior, consistently achieving faster convergence, consistent numerical stability, and higher accuracy. Consequently, there may not be any need for the use of explicit models except there is a problem of numerical instability from the use of the Colebrook equation. In this study, however, the Colebrook equation, solved using the Clamond method did not exhibit any numerical instability across all four network case studies.

References

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