Hardy Cross Method Research Articles

Analysis of Water Distribution Network Using EPANET

The main purpose of the water distribution network is to supply water at adequate pressure and flow in the performance of urban areas and potable quality of water to the consumers. This paper highlights the effective analysis and distribution of network of pipes using EPANET tool as well as the Hardy Cross method and Newton-Raphson method. EPANET is the computer program that performs the extended period simulation of hydraulic behaviour within pressurized pipe network. A simulation will be carried out for hydraulic parameters such as pressure and flow rate. The results will be verified all junctions and the flows with their velocities in all pipes feasible enough to provide adequate water to the network of the study area. The residual head at each and every node will be found out by knowing the elevation as input and thereby the corresponding flow parameters will be derived like residual head, velocity, and nodal demand etc. The findings may help to understand the pipelines system of the study area in a better way. This work deals with the analysis of urban water distribution network in developing countries.

An Efficient Iterative Method for Looped Pipe Network Hydraulics Free of Flow-Corrections

The original and improved versions of the Hardy Cross iterative method with related modifications are today widely used for the calculation of fluid flow through conduits in loop-like distribution networks of pipes with known node fluid consumptions. Fluid in these networks is usually natural gas for distribution in municipalities, water in waterworks or hot water in district heating systems, air in ventilation systems in buildings and mines, etc. Since the resistances in these networks depend on flow, the problem is not linear like in electrical circuits, and an iterative procedure must be used. In both versions of the Hardy Cross method, in the original and in the improved one, the initial result of calculations in the iteration procedure is not flow, but rather a correction of flow. Unfortunately, these corrections should be added to or subtracted from flow calculated in the previous iteration according to complicated algebraic rules. Unlike the Hardy Cross method, which requires complicated formulas for flow corrections, the new Node-loop method does not need these corrections, as flow is computed directly. This is the main advantage of the new Node-loop method, as the number of iterations is the same as in the modified Hardy Cross method. Consequently, a complex algebraic scheme for the sign of the flow correction is avoided, while the final results remain accurate.

Evaluation of Flow Rate Correction in Water Pipeline Distribution Network by Two Numerical Methods of Solution

This study evaluates flow rate correction and approximate flow rates in loops for three different case studies of closed looped pipe distribution network systems using Hardy Cross and Newton Raphson. Darcy Weisbach head loss equation was also used to account for major losses. Manual calculation was initially done for each case study followed by a C-Sharp programming software which was developed to affirm the manual calculation. For one looped network, head loss around the loop converged from 25.60 m to 0.13 m at the third iteration. The two looped network head loss around each loop converged from 170.97 m and 8.92 m to 0.05 and 0.06 m for Hardy Cross at the sixth iteration while the head loss are 0.88 m and 0.24 m at the fourth iteration for both Hardy Cross and Newton Raphson method while for the three looped network, it has head losses around the three loops converged after the fourth iteration from 0.26, 1.36 and 18.32 m to 0.13, 0.11 and 0.10 m respectively for Hardy Cross at third iteration while the head losses are 0.03, 0.00 and 0.05 m for Newton Raphson method. Newton Raphson method was found to have a better convergence pattern because it convergences in a uniform manner unlike Hardy Cross method. Also, the program developed gave almost but more accurate results as compared to that of manual calculations with the agreement between them rated at 98%. Some slight differences encountered in the mathematical terms calculated were as a result of some accumulated approximation errors. Keywords : Pipe Distribution Network, Head loss, Convergence, Iteration

Stokes Flow Within Networks of Flow Branches

Stokes flow in the branches of structured looped networks with successive identical square loops and T-junction branches is studied. Analytical expressions of the flow rate in the branches are determined for network of one, two, three, or four loops with junction head loss neglected relative to regular one. Then, a general expression of the flow rate is deduced for networks with more loops. This expression contains particularly a sequence of coefficients obeying to a recurrence formula. This sequence is a part of the fusion of Fibonacci and Tribonacci sequences. Furthermore, a general formula that expresses the quotient of flow rate in successive junction flow branches is given. The limit of this quotient for an infinite number of junction branches is found to be equal to 2+3. When the inlet and outlet flow rates are equal, this quotient obeys to a sequence of invariant numbers whatever the ratio of flow rate in the outlet branches is. Thus, the flow rate distribution for any configuration of inlet and outlet flow rates can be calculated. This study is also performed using Hardy–Cross method and a commercial solver of Navier-Stokes equation. The analytical results are approached very well with Hardy–Cross method. The numerical resolution agrees also with analytical results. However, the difference with the numerical results becomes significant for low flow rate in the junction branches. The flow streamlines are then determined for some inlet and outlet flow rate configurations. They particularly illustrate that recirculation flow takes place in branches of low flow rate.

Analisis Sistem Distribusi Air Bersih di Komplek Perumahan Cemara Hijau Medan

Cemara Hijau Housing Complex is an elite housing complex located on the outskirts of Medan which each year has an increasing number of consumers not followed by increased network capacity and clean water supply. This study aims to analyze the pattern of water consumption of customers in the Green Pine complex, analyzing the availability of PDAM Tirtanadi water of Cemara branches with water needs of Cemara Hijau customers and analyzing the clean water distribution system in Cemara Hijau housing using Hardy Cross method with Hazen William and Epanet 2.0 equations. The research method used is to first conduct literature studies related to clean water distribution system and Epanet 2.0 application. Then collect the required data that is primary data and secondary data. Then the data data is analyzed by manual calculation of Hardy Cross with Hazen William equation and using computer software that is Epanet 2.0. The result of research is the pattern of clean water consumption in the green pine complex with the highest factor is 1.69 while the lowest is 0.21, then the water loss at the Cemara Hijau Housing Complex is 52.93%, then the clean water distribution network system in the Cemara Hijau Housing Complex does not meet in terms of flow and pressure, so that the solution given is the addition of 50 mm diameter made in parallel, the last of the total difference obtained from Epanet 2.0 with a large enough field for the hourly of 0.0766 m3/sec and for 24 hours of 4.3878 m3/sec, and the total difference between Epanet 2.0 with Hardy Cross method with Hazen William equation is not too large with a total difference of 0.00048 m3/sec or 0.48 liter/sec.

Automated Framework for Water Looped Network Equilibrium

In this paper, a novel algorithm is proposed for balancing water looped network in steady state through a fully automated general framework of hydraulic networks regardless of their topological complexity. The model is developed by combining the following two steps, firstly a set of independent loops are identified based on a graph theoretical analysis in a looped network. Further the second step is devoted to the equilibrium process by determining the flow rate distribution within the network ducts and the pressure in the delivery nodes. The above such equilibrium process gives rise to a system of non linear algebraic equations which are solved numerically using both Hardy Cross (HC) and Newton Raphson (NR) methods. In HC method, the flow correction term is modified and a generalized expression is given to consider various possibilities of independent loops selection. Some real networks topologies that were commonly used as benchmarks, for testing various independent loops selection algorithms, are taken as case studies to apply the general automatic framework for hydraulic network analysis. Such network analysis enhances proving the applicability as well as the effectiveness of the proposed approach. Also, during the equilibrium procedure, it is proved that NR method is capable of producing accurate results and it converges more rapidly comparing to the widely used HC method. Moreover, it is demonstrated that NR’s iterative process, contrary to HC’s one, converges to reliable results even with a choice of random initial flow rates which makes a NR algorithm quite simple to implement without affecting the accuracy of the results.

Numerical modelling of hydrogen-natural gas mixtures flows in looped networks

This article concerns the numerical analysis of high pressure hydrogen-natural gas mixtures flows in pipeline networks during steady and transient states. The considered fluid is an homogeneous mixture of hydrogen and natural gas. An isentropic process is admitted for both components and under such assumption the density of the binary gas mixture is defined. The steady state was studied by the use of Hardy Cross method. The numerical simulation of the transient regime was performed by solving the conservation equations, for one-dimensional isentropic compressible flow, using the characteristics method of specified time intervals. The obtained results have proved the efficiency of the characteristics method compared to other numerical techniques. The numerical obtained results have shown that, during transients, pressure oscillations for hydrogen and hydrogen-natural gas mixtures are higher compared to those for natural gas.

네트워크형 터널의 환기해석 방법에 대한 기초연구-비 Hardy-Cross 방법을 중심으로

Recently, various forms of diverging sections in tunnels have been designed as the demand for underground passageway in urban areas increases. Therefore, the complexity of the ventilation system in tunnels with diverging sections requires a ventilation analysis method different from the conventional method for the straight tunnels. None of the domestic and foreign tunnel ventilation design standards specifies the method for the ventilation network analysis, and the numerical analysis methods have been most widely used. This paper aims at reviewing the ventilation network analytical method applicable as the design standard. The proposed method is based on the characteristic equations rather than the numerical analysis. Thanks to the advantages of easy application, the Hardy-Cross method has been widely applied in the fields of mine ventilation and tunnel ventilation. However, limitations with the cutting errors in the Taylor series expansion and the convergence problem mainly caused by the mesh selection algorithm have been reported. Therefore, this paper examines the applicability of the ventilation analysis method for network-type tunnels with the gradient method that can analyze flow rate and pressure simultaneously without the configuration of mesh. A simple ventilation analysis method for network-type tunnels is proposed.

A general characterization of the Hardy Cross method as sequential and multiprocess algorithms

The Hardy Cross method of moment distribution admits, for any problem, an entire family of distribution sequences. Intuitively, the method involves clamping the joints of beams and frames against rotation and balancing moments iteratively, whether consecutively, simultaneously, or in some combination of the two. We present common versions of the moment distribution algorithm and generalizations of them as both sequential and multiprocess algorithms, with the latter exhibiting the full range of asynchronous behavior allowed by the method. We prove, in the limit, that processes so defined converge to the same unique solution regardless of the distribution sequence or interleaving of steps. In defining the algorithms, we avoid overspecifying the order of computation initially using a sequential, nondeterministic process, and then more generally using concurrent processes.

Designing of Hydraulically Balanced Water Distribution Network Based on GIS and EPANET

The main objectives of this paper are, designing and balancing of Water Distribution Network (WDN) based on loops hydraulically balanced method as well as using Geographical Information System (GIS) methodology with the contribution of EPANET. GIS methodology is used to ensure WDN’s integrity and skeletonized a proper and functional WDN by using Network Analyst utilizing the geometric network and topology network by hierarchical geo-databases. The problem is to make WDN hydraulically balanced by applying WDN balancing method. For that reason, we have analyzed water flows in each pipe and performed the iterations process on loops in order to make the algebraic summation of head loss“h_f” around any closed loop zero. In case, the summation of pipe flows must be equal to the flow amount entering or leaving the system through each node. At each iteration, reasonable changes occurred at pipes flow until the head loss has become very small or fixed zero as (optimizes correction) by using excel sheet solver. Since this method is confirmed to be effective, simulations were done by using GIS and EPANET water distribution platform. As a result, we accomplished hydraulically balanced WDN. Finally, we have analyzed and simulated hydraulics parameters for the targeted area in Kabul city. Thus, determined successfully the hydraulics state of parameters around the network as a positive result. It is worth mentioning that, Hardy-cross method is being used for approaching more precise optimized correction and consequences concerning hydraulically-balanced and optimal WDN. This method can be done for complex loops WDN as well; the advantage of the method is simple math and self-correction. Managers and engineers who work in the field of water supply this methodology has been recommended as the more advantageous workflow in planning water distribution pattern.

Analysis of a Looped High Pressure Steam Pipeline Network in a Large-Scale Refinery

This work presents a rigorous analytical method to evaluate a looped high pressure steam distribution network in an oil refinery. The Hardy Cross method is modified to handle the nonisothermal and compressible flow of high pressure steam in the looped pipeline network. Steam conditions taken into account include the flow pattern, the pressure, and temperature profiles. The amount of condensate generated in pipeline sections in the looped network is calculated with high precision. The movement of possible stagnant points, pipe sections where steam flow rate is significantly small, is also identified. Scenario-based analyses indicate that steam temperature drop is significant in those sections with stagnant points. An oil refinery can benefit greatly from using the proposed analysis to optimize operation of a high pressure steam distribution network.

Forecasting semi-dynamic response of natural gas networks to nodal gas consumptions using genetic fuzzy systems

Abstract-Semi-dynamic behavior of natural gas distribution network and nodal gas consumptions are predicted. Traditional Hardy-Cross method for analysis of the gas network is replaced with a direct mathematical solution of mass conservation equations at network nodes to yield nodal static pressures and volumetric flow rates for the coming days. After the calculation of static pressure distribution in a network for near future days, the problem of pressure drop in the network which is a serious problem in cold seasons can be managed in advance. TSK (Takagi-Sugeno-Kang) fuzzy system is used for forecasting. Structure identification of the system is carried out using CVIs (Cluster Validity Indices) and PFCM (Possibilistic Fuzzy C-Means algorithm) to determine number of rules which is also chosen such that testing error of the system does not exceed a predefined value. Premise and t-norm parameters of the TSK system are tuned by GAs (Genetic Algorithms) and their consequent parameters are adjusted using LSE (Least Square Estimate). Comparison of testing error of the TSK system for modeling benchmark data with other popular methods demonstrates its suitability for forecasting nodal gas consumptions.

Hydraulic Analysis of Water Supply Networks Using a Modified Hardy Cross Method

There are different methods for the hydraulic analysis of water supply networks. In the solution process of most of these methods, a large system of linear equations is solved in each iteration. This usually requires a high computational effort. Hardy Cross method is one of the approaches that do not need such a process and may converge to the solution through scalar divisions. However, this method has two shortcomings: first, initial discharges should satisfy continuity equation at each node; second a large number of iterations are required to converge to solution. In this article an algorithm is suggested for initial discharges that are close to the final results while the continuity equations are automatically established. This algorithm may be directly implemented in the Hardy Cross method. To reduce the number of iterations the Hardy Cross method is combined with third-order and sixteenth-order methods. The results of some numerical examples demonstrate that the use of the combined approach with the suggested initial guess reduces the number of iterations and hydraulic analysis time and the solutions converge with a high accuracy.

Flow rate distribution and effect of convection and radiation heat transfer on the temperature profile during a coil annealing process

Determining the temperature of several steel coils, heated in a furnace with a controlled hydrogen environment is important in an annealing process. Temperatures must be defined during heat treatment in order to guarantee metallurgical properties and acceptable reduced residual stresses. In this paper we approach hydrogen flow characteristics in the furnace and through a set of coils using an annealing non-differential model. Fluid flow is schematized as a pipe network solved by the Hardy Cross method to obtain pressure drops across the various gas flow segments. A comparison is made between measured and simulated results, confirming the adequacy of adopted assumptions and the validity of proposed model. Convective and radiative exchanges between the furnace and the coils are calculated by a discretization using the finite differences method. The convection coefficients are estimated and introduced into the boundary conditions around the coil to obtain the temperature distribution in the coils and in the covering bell. Finally, heat exchanges by convection and radiation are estimated by this model and the modeling errors are <8 °C.

Nodal Analysis of Urban Water Distribution Networks

There are three methods for analysing the flow and pressure distribution in looped water supply networks (the loop method, the node method, the pipe method), accounting for the chosen unknown hydraulic parameters. For all of these methods, the nonlinear system of equations can be solved using iterative procedures (Hardy–Cross, Newton–Raphson, linear theory). In the cases of the extension or the rehabilitation of distribution networks, the unknown parameters are the hydraulic heads at nodes, and the nodal method for network analysis is preferred. In this paper, a generalised classic model is developed for the nodal analysis of complex looped systems with non-standard network components and the solvability of new problems, along with the determination of the pressure state in the system. In addition, this paper exhibits a different approach to this problem by using the variational formulation method for the development of a new analysis model based on unconditioned optimisation techniques. This model has the advantage of using a specialised optimisation algorithm, which directly minimises an objective multivariable function without constraints, implemented in a computer program. The two proposed models are compared with the classic Hardy–Cross method, and the results indicated a good performance of these models. Finally, a study is performed regarding the implications of the long-term operation of the pipe network on energy consumption using these models. The new models can serve as guidelines to supplement existing procedures of network analysis.

Gröbner bases for solving ΔQ-equations in water distribution networks

The analysis of water distribution network is of great interest to hydraulic engineers. Although the water distribution network has been extensively studied for the last decades, there are still many unsolved problems awaiting clarification. In this paper, an algorithm is presented that describes a computationally efficient technique for water distribution networks based on Gröbner basis method. Gröbner basis algorithm provides the exact algorithmic solutions for solving the system of equations. However, Gröbner algorithm works only for polynomials and moreover for a large scale network, it takes a long CPU time. Hence, we present two other algorithms that work for non-polynomials and large scale problems. Three examples are presented to show the effectiveness of Gröbner basis method compared with Hardy Cross method, linear theory and Gradient method.

Network Analysis of Nonuniformly Heated Evaporative Microchannel Systems

A network analysis was established to model the array of evaporative microchannels with possible nonuniformity heating as well as branching of the channels. Iterative solution of the evaporative microchannel network can be obtained using the Hardy–Cross method together with accurate two-phase head-loss correlations. Based on the experimental evidence, cross-cutting microchannels reduce the uneven flow distribution in parallel microchannels at nonuniform heating. Through this network analysis, it is also evident that cross-cutting grooves on parallel microchannels are effective in reducing nonuniform heating effects and enhancing the uniform wall temperature distribution.

A Gas Distribution Network Hydraulic Problem from Practice

The purpose of this is the determination of appropriate friction factor and selection of a representative equation for natural gas flow under the presented conditions in the network. Calculation of the presented looped gas pipeline network is done according to principles of Hardy Cross method. The final flows were calculated for known pipe diameters and nodes consumptions while the flow velocities through pipes have to stand below certain values. In optimization problems flows are treated as constant, and the diameters are variables.

Study on heat transfer and pressure drop characteristics of internal heat exchangers in CO2 system under cooling condition

In order to study the heat transfer and pressure drop on four types of internal heat exchangers (IHXs) of a CO2 system, the experiment and numerical analysis were performed under a cooling condition. The configuration of the IHXs was a coaxial type and a micro-channel type. Two loops on the gas cooler part and the evaporator part were made, for experiment. And the section-by-section method and Hardy-Cross method were used for the numerical analysis. The capacity and pressure drop of the IHX are larger at the micro-channel type than at the coaxial type. When increasing the mass flow rate and the IHX length the capacity and pressure drop increase. The pressure drop of the evaporator loop is much larger than that of the gas cooler loop. The performance of the IHX was affected with operating condition of the gas-cooler and evaporator. The deviations between the experimental result and the numerical result are about ±20% for the micro-channel type and ±10% for the coaxial type. Thus, the new CO2 heat transfer correlation should be developed to precisely predict a CO2 heat transfer.

An improvement of Hardy Cross method applied on looped spatial natural gas distribution networks

Hardy Cross method is common for calculation of loops-like gas distribution networks with known node gas consumptions. This method is given in two forms: original Hardy Cross method-successive substitution methods and improved-simultaneous solution method (Newton–Raphson group of methods). Problem of gas flow in looped network is nonlinear problem; i.e. relation between flow and pressure drop is not linear while relation between electric current and voltage is. Improvement of original method is done by introduction of influence of adjacent contours in Yacobian matrix which is used in calculation and which is in original method strictly diagonal with all zeros in non-diagonal terms. In that way necessary number of iteration in calculations is decreased. If during the design of gas network with loops is anticipated that some of conduits are crossing each other without connection, this sort of network became, so there has to be introduced corrections of third or higher order.