User:Tater2650/Hardy Cross Method

The Hardy Cross Method, or “The Relaxation Method” was developed by the American Engineer Hardy Cross and is still being used today to help design water supply networks and in mine ventilation networks. This method helps correct assumed flows when circuit pressure drops occur in the system and therefore cause a need to balance the circuit pressure drops. By definition, in any closed circuit the pressure drop has to be zero, to fix this error the flow in each of the loops must be corrected until the resulting error is within the allowable tolerance for the system. The Hardy Cross Method should not be confused with the moment distribution method which was developed by Hardy Cross as well.

The idea that any water flow network can be simplified by incorporating the minimum number of independent circuits. The minimum required number of independent circuits can be directly found by calculating the minimum spanning tree. Then the basic edges which don’t form part of the minimum spanning tree are eliminated.

Procedure
This procedure for the Hardy Cross Method can be used to analyze many different pipe networks ranging for the most simple to very complex. The following is for analyzing a simple pipe network.

Steps

 * 1) Locate and number each of the different loops within the system.
 * 2) Assume the following flow directions, clockwise is positive and counterclockwise is negative.
 * 3) Assume an initial flow through each individual pipe.
 * 4) Using the same sign convention as the flow directions, calculate the head loss in each of the unique loops.
 * 5) Check the loops closure by adding up the head losses of the individual pipes in the loop.
 * 6) Calculate flow corrections to improve head loss
 * 7) Repeat the process until the head losses and closures are within the acceptable tolerance for the system.

Additional Procedural Information
Simulations for Pipe Networks help demonstrate the steady flow of pressurized gases or liquids and can be used to simulate a wide range of areas including but not limited to city water systems and car exhaust manifolds. These real life systems include long pipelines and tubes (relative to the size of the system) with different diameters of tubing and piping in parallel or series and in some large scale systems, mainly waste water treatment plants, water will flow into slotted screens, extraction units, clarifiers and aerators. As stated earlier in all these systems the sum of all the in flowing fluid must equal the sum of all the liquid flowing out, and based on one input pressure all other pressures can be calculated. Then once the Hardy Cross Method is applied one can eliminate superfluous pipes and tubes.

The Hardy Cross Method takes consecutive flow estimations based on the following two conditions; At any junction between any number of pipes the sum of the flow into the junction must equal the sum of the flow out of the junction, or ΣQ= 0 and between any two junctions, the total head loss is independent of the path taken based on the conservation of energy. In a network with “N” junctions, exactly “(N-1)” equations can be formed from these junctions along with the estimated flows for the respective junctions to help determine the flows throughout the system. The information required to apply the Hardy Cross Method includes: The estimated flows in each of the included pipes, the diameters of every pipe in the system, the length of each pipe, the friction coefficients, the connectivity, and junction elevations. Once applied the Hardy Cross Method will result in the head loss for all the pipes in the system based on the estimated flow rates decided upon earlier. Because all flow rates are assumed it is expected that the loss of head in the clockwise direction will equal the loss of head in the counterclockwise direction and the difference between the two is known as closure error, the main goal of the designer is to get the closure error down to 0 or within an acceptable range of values for the given problem or system.

There are many variations to the Hardy Cross Method including: the General Method or The Equivalent Pipes technique, The Alternative Equivalent Pipe Procedure. In the General Method pipes are removed from a system when they are not needed because there is no flow into them. The equivalent pipe technique involves the exchange of one pipe of both a certain diameter and a variable length or vice versa and having a variable diameter and a set length. As long as the system doesn’t add more or remove fluid from the network this technique will hold true. As long as the condition is true the pipes are exchanged to allow for a more efficient flow. In the Alternative equivalent pipe procedure is the process involved when a network of pipes is converted to one pipe to keep the loss of head low. The more complex the system of pipes with multiple input and output nodes the greater the chance is that another method from the Hardy Cross will have to be used because as stated earlier the Hardy Cross Method requires no input or output between the end nodes in the system. However on single input output systems, the Hardy Cross method is the best option to reach hydraulic equilibrium.

There are also some procedures and techniques that are similar to the Hardy Cross Method, the most common of these methods is linear programming which can be used for a variety of simple water flow models, traffic flow models, and other similar flow type problems. It is similar in the fact that it follows the same principal that whatever fluid goes in a node must come back out. The main difference is that usually flows are not generalized but rather they are capped to a maximum value and constrained so that they flow in one direction (which is usually assumed in Hardy Cross applications). Other variations of linear programming include Quadratic programming (which uses quadratic equations as opposed to linear models to model flows), and integer programming ( identical in method and approach to linear modeling but is also constrained to only allow for integer outputs in the system and can also be used to model product lines).

History
Hardy Cross first developed the moment distribution method which was to be used to calculate within a required degree of accuracy by successive approximations while being able to avoid the immense task of solving multiple equations simultaneously  and is also called The Hardy Cross Method. In his later work, Hardy Cross adapted his research and developed his trial and error method for water pipe networks during the year 1936, which was later improved by Fair Hurst and G.M. Fair Howland. Cross’s work on water pipelines was later applied to solving similar flow network problems such as gas pipelines and municipal water supply designs. Today Hardy Cross’s work is still being used but now with the advances in technology and increasing complexity of cities and their accompanying water systems the analysis is done by computers quickly and efficiently unless the system is very small relative to the modern standard.

Current Uses
Taiwanese Municipal Water System 2003 Currently the Hardy Cross Method is being applied to many different fluid flow systems very recently it has been applied to help with Taiwan’s municipal water supply systems. By applying the Hardy Cross Method researchers Ming-Chang Tsai, Ying-Hui Chang and Tien-Yin Chou from Feng Chia University in Taichung Taiwan have been able to find each water sources supply volume in the major system. Due to the rapid development in the Taiwanese municipal water industry, water had to be pumped from rivers by being dammed and then directly piped and purified in a special facility. The process turned the municipal water into both a technical and service type of industrial product as well as raising the local Taiwanese people’s standard of living and improved social welfare. The study involved the research and application in geographic information system technologies, and the combination of geographic information and pipe network water pressure analysis method and conformed special information in pipe network water pressure analysis processes. The required geographic information is found using graph theory and formed from one unit connected to a point with a line segment. After all the analysis was done in Taiwan, the researchers concluded that with a traditional network analysis that considered all the pipe segments to have equal qualities and that they all carried and dispersed the same amount of water, would give an unacceptable amount of municipal water to the people of Taiwan. The solution to the problem as the use larger pipes near the sources of the water and as it flowed nearer to its destination to begin using smaller pipes to distribute the water among the people.

Common Analysis
Other common analysis techniques used by many different engineers and other professionals alike which are very similar to the Hardy Cross Method are the steady state analysis and the time dependent analysis on Microsoft Excel which using a very simple algorithm can calculate the flows of anything from fluids to traffic in a system that has different nodes (Pipe and road intersections). This analysis on Excel is very useful because if set up correctly it can save the researchers the time required for the traditional Hardy Cross Method because of Excel’s built in Solver function. This technique is easily applicable to education in flow systems and in the professional world in many different applications.