Short overview of early developments of the Hardy Cross type methods for computation of flow distribution in pipe networks

Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before the Hardy Cross method. A problem with flow resistance versus electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties, and it remains an important tool for the analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method that considers the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from a Russian practice published during the 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the work of Hardy Cross are also presented. Finally, an improved version of the Hardy Cross method, which significantly reduces the number of iterations, is presented and discussed. We also tested multi-point iterative methods, which can be used as a substitution for the Newton-Raphson approach used by Hardy Cross, but in this case this approach did not reduce the number of iterations. Although many new models have been developed since the time of Hardy Cross, the main purpose of this paper is to illustrate the very beginning of modeling of gas and water pipe networks and ventilation systems. As a novelty, a new multi-point iterative solver is introduced and compared with the standard Newton-Raphson iterative method.Web of Science910art. no. 201

Steady-State Analysis of Electrical Networks in Pandapower Software: Computational Performances of Newton–Raphson, Newton–Raphson with Iwamoto Multiplier, and Gauss– Seidel Methods

At the core of every system for the efficient control of the network steady-state operation is the AC-power-flow problem solver. For local distribution networks to continue to operate effectively, it is necessary to use the most powerful and numerically stable AC-power-flow problem solvers within the software that controls the power flows in these networks. This communication presents the results of analyses of the computational performance and stability of three methods for solving the AC-power-flow problem. Specifically, this communication compares the robustness and speed of execution of the Gauss–Seidel (G–S), Newton–Raphson (N–R), and Newton–Raphson method with Iwamoto multipliers (N–R–I), which were tested in open-source pandapower software using a meshed electrical network model of various topologies. The test results show that the pandapower implementations of the N–R method and the N–R–I method are significantly more robust and faster than the G–S method, regardless of the network topology. In addition, a generalized Python interface between the pandapower and the SciPy package was implemented and tested, and results show that the hybrid Powell, Levenberg–Marquardt, and Krylov methods, a quasilinearization algorithm, and the continuous Newton method can sometimes achieve better results than the classical N–R method

Application of Numerical Newton-Raphson Method in Calculation of Emitter Water Discharge of Drip Irrigation System in “Mutis Cemerlang” Coffee Plantation

Based on the slope class, the shape of the area and the elevation of dry land in Nusa Tenggara, agricultural land in Noepesu Village is suitable for planting coffee plants with an agroforestry scheme. To overcome the problem of limited water, drip irrigation system technology can be applied. The use of drip irrigation as an agricultural technology in Noepesu village has been carried out by many farmer groups. Still, the installation process does not consider the pipe specifications (pipe length and pipe diameter) and the condition of agricultural land. This causes the service life of drip irrigation to be not long. If this continues, of course, it will increase system installation costs. To optimize service life, a hydraulics analysis method is needed for drip irrigation pipe network systems that take into account pipe specifications and agricultural land conditions.The hydraulics analysis of the drip irrigation network system determines the emitter’s water flow rate. The emitter flow rate forms a nonlinear equation known as the closed pipe equation. In the process of solving these equations, numerical methods can be used, specifically the Newton-Raphson method. This study focuses on applying the Newton-Raphson method to calculate the amount of water discharge from each emitter of the drip irrigation network system on the farmland of the Mutis Cemerlang Farmer Group in Noepesu Village. The drip irrigation system is designed with 250 nodes, 275 pipes, 26 loops, and 86 outlets divided into two sides, with the left side containing 84 outlets with one emitter and the right side containing 102 outlets with two emitters. The amount of water discharge for each emitter is 0,0008 ml/second≤Q≤2,6 ml/second for the left side and 0,001 ml/second≤Q≤1,1 ml/second for the right side, as determined by simulation calculations utilizing the Newton-Raphson method and Matlab software. The simulation results show that the amount of water discharge at each emitter is ideal in the first iteration because it has a discharge correction value (∆Q)≈0

Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function: Reply to Discussion

This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples

What can students learn while solving Colebrook's flow friction equation?

Even a relatively simple equation such as Colebrook's offers a lot of possibilities to students to increase their computational skills. The Colebrook's equation is implicit in the flow friction factor and, therefore, it needs to be solved iteratively or using explicit approximations, which need to be developed using different approaches. Various procedures can be used for iterative methods, such as single the fixed-point iterative method, Newton-Raphson, and other types of multi-point iterative methods, iterative methods in a combination with Pade polynomials, special functions such as Lambert W, artificial intelligence such as neural networks, etc. In addition, to develop explicit approximations or to improve their accuracy, regression analysis, genetic algorithms, and curve fitting techniques can be used too. In this learning numerical exercise, a few numerical examples will be shown along with the explanation of the estimated pedagogical impact for university students. Students can see what the difference is between the classical vs. floating-point algebra used in computers.Web of Science43art. no. 11

Advanced modelling and simulation of water distribution systems with discontinuous control elements

Water distribution systems are large and complex structures. Hence, their construction, management and improvements are time consuming and expensive. But nearly all the optimisation methods, whether aimed at design or operation, suffer from the need for simulation models necessary to evaluate the performance of solutions to the problem. These simulation models, however, are increasing in size and complexity, and especially for operational control purposes, where there is a need to regularly update the control strategy to account for the fluctuations in demands, the combination of a hydraulic simulation model and optimisation is likely to be computationally excessive for all but the simplest of networks. The work presented in this thesis has been motivated by the need for reduced, whilst at the same time appropriately accurate, models to replicate the complex and nonlinear nature of water distribution systems in order to optimise their operation. This thesis attempts to establish the ground rules to form an underpinning basis for the formulation and subsequent evaluation of such models. Part I of this thesis introduces some of the modelling, simulation and optimisation problems currently faced by water industry. A case study is given to emphasise one particular subject, namely reduction of water distribution system models. A systematic research resulted in development of a new methodology which encapsulate not only the system mass balance but also the system energy distribution within the model reduction process. The methodology incorporates the energy audits concepts into the model reduction algorithm allowing the preservation of the original model energy distribution by imposing new pressure constraints in the reduced model. The appropriateness of the new methodology is illustrated on the theoretical and industrial case studies. Outcomes from these studies demonstrate that the new extension to the model reduction technique can simplify the inherent complexity of water networks while preserving the completeness of original information. An underlying premise which forms a common thread running through the thesis, linking Parts I and II, is in recognition of the need for the more efficient paradigm to model and simulate water networks; effectively accounting for the discontinuous behaviour exhibited by water network components. Motivated largely by the potential of contemplating a new paradigm to water distribution system modelling and simulation, a further major research area, which forms the basis of Part II, leads to a study of the discrete event specification formalism and quantised state systems to formulate a framework within which water distribution systems can be modelled and simulated. In contrast to the classic time-slicing simulators, depending on the numerical integration algorithms, the quantisation of system states would allow accounting for the discontinuities exhibited by control elements in a more efficient manner, and thereby, offer a significant increase in speed of the simulation of water network models. The proposed approach is evaluated on a number of case studies and compared with results obtained from the Epanet2 simulator and OpenModelica. Although the current state-of-art of the simulation tools utilising the quantised state systems do not allow to fully exploit their potential, the results from comparison demonstrate that, if the second or third order quantised-based integrations are used, the quantised state systems approach can outperform the conventional water network simulation methods in terms of simulation accuracy and run-time

Mass transfer of solutes in turbulent wall bounded flows reacting with the conduit surface

This thesis focuses on the decay of chlorine in pipes of drinking water distribution networks due to wall and bulk demand. Accurate prediction of chlorine decay is important, as both chlorine concentrations which are too low and too high pose serious health risks, the former due to pathogen formation and the latter due to the formation of disinfection by-products. Water quality models used for the prediction of chlorine decay make use of parameterisations for the wall demand in the form of Sherwood number Sh correlations, which couple the wall mass flux to a Reynolds number Re, Schmidt number Sc and wall roughness. These correlations are subject to significant uncertainty, particularly for turbulent flows. A combined analytical and numerical approach is taken to study in detail the interaction between flow, turbulence and mass transport, with the aim of improving the understanding and accuracy of wall demand parameterisations for chlorine. Simulations of the chlorine decay in an axisymmetric pipe with hydraulically smooth walls were performed for Re = 104 to 106 and Sc = 1000 using Reynolds averaged conservation equations. These values are typical for chlorine transport in distribution networks. The simulations confirmed that the assumptions made in water quality models for chlorine wall demand are valid. Asymptotic solutions for high Sc solutes were developed which are applicable both to linear and nonlinear wall reactions. Results showed that the Sh correlation is independent of the reaction type. For rough walls, the two main wall demand parameterisations are mutually inconsistent: one is valid for low and the other for high wall demand coefficients only. Numerical simulation of flow and high Sc mass transport over a dtype rough surface at Re = 2.5×105 showed that the inconsistency between the two parameterisations was caused by the geometry. For low wall demand coefficients, the existence of roughness elements causes higher wall demand than for a smooth wall. However, at high wall demand coefficients the maximum wall demand achievable in the cavities was much smaller than for the crests. Hence, the effective surface area and therefore the wall demand became lower than for a smooth wall. A parameterisation was developed which reproduced the solute mass decay over the entire range of wall demand coefficients. Most of the solutions and parameterisations developed in this thesis are on the same level of description as water quality models. The findings of this thesis can be used as supportive evidence for the validity of assumptions made for water quality models, and to inform how processes should be modelled when these assumptions are violated

SEGMENT-BASED RELIABILITY ASSESSMENT FOR WATER DISTRIBUTION NETWORKS

In recent years, water utilities have placed a greater emphasis on the reliability and resilience of their water distribution networks. This focus has increased due to the continuing aging of such infrastructure and the potential threat of natural or man-made disruptions. As a result, water utilities continue to look for ways to evaluate the resiliency of their systems with a goal of identifying critical elements that need to be reinforced or replaced. The simulation of pipe breaks in water reliability studies is traditionally modeled as the loss of a single pipe element. This assumes that each pipe has an isolation valve on both ends of the pipe that can be readily located and operated under emergency conditions. This is seldom the case. The proposed methodology takes into account that multiple pipes may be impacted during a single failure as a result of the necessity to close multiple isolation valves in order to isolate the “segment” of pipes necessary to contain the leak. This document presents a simple graphical metric for use in evaluating the performance of a system in response to a pipe failure. The metrics are applied to three different water distribution systems in an attempt to illustrate the fact that different pipe segments may impact system performance in different ways. This information is critical for use by system managers in deciding which segments to prioritize for upgrades or replacement