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fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
, pipe network analysis is the analysis of the
fluid flow In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
through a
hydraulics Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counter ...
network, containing several or many interconnected branches. The aim is to determine the flow rates and
pressure drop Pressure drop is defined as the difference in total pressure between two points of a fluid carrying network. A pressure drop occurs when frictional forces, caused by the resistance to flow, act on a fluid as it flows through the tube. The main det ...
s in the individual sections of the network. This is a common problem in hydraulic design.


Description

To direct water to many users, municipal water supplies often route it through a
water supply network A water supply network or water supply system is a system of engineered hydrologic and hydraulic components that provide water supply. A water supply system typically includes the following: # A drainage basin (see water purification – source ...
. A major part of this network will consist of interconnected pipes. This network creates a special class of problems in hydraulic design, with solution methods typically referred to as ''pipe network analysis''. Water utilities generally make use of specialized software to automatically solve these problems. However, many such problems can also be addressed with simpler methods, like a spreadsheet equipped with a solver, or a modern graphing calculator.


Deterministic network analysis

Once the friction factors of the pipes are obtained (or calculated from pipe friction laws such as the Darcy-Weisbach equation), we can consider how to calculate the flow rates and head losses on the network. Generally the head losses (potential differences) at each node are neglected, and a solution is sought for the steady-state flows on the network, taking into account the pipe specifications (lengths and diameters), pipe friction properties and known flow rates or head losses. The steady-state flows on the network must satisfy two conditions: # At any junction, the total flow into a junction equals the total flow out of that junction (law of conservation of mass, or continuity law, or Kirchhoff's first law) # Between any two junctions, the head loss is independent of the path taken (law of conservation of energy, or Kirchhoff's second law). This is equivalent mathematically to the statement that on any closed loop in the network, the head loss around the loop must vanish. If there are sufficient known flow rates, so that the system of equations given by (1) and (2) above is closed (number of unknowns = number of equations), then a ''deterministic'' solution can be obtained. The classical approach for solving these networks is to use the Hardy Cross method. In this formulation, first you go through and create guess values for the flows in the network. The flows are expressed via the volumetric flow rates Q. The initial guesses for the Q values must satisfy the Kirchhoff laws (1). That is, if Q7 enters a junction and Q6 and Q4 leave the same junction, then the initial guess must satisfy Q7 = Q6 + Q4. After the initial guess is made, then, a loop is considered so that we can evaluate our second condition. Given a starting node, we work our way around the loop in a clockwise fashion, as illustrated by Loop 1. We add up the head losses according to the Darcy–Weisbach equation for each pipe if Q is in the same direction as our loop like Q1, and subtract the head loss if the flow is in the reverse direction, like Q4. In other words, we add the head losses around the loop in the direction of the loop; depending on whether the flow is with or against the loop, some pipes will have head losses and some will have head gains (negative losses). To satisfy the Kirchhoff's second laws (2), we should end up with 0 about each loop at the steady-state solution. If the actual sum of our head loss is not equal to 0, then we will adjust all the flows in the loop by an amount given by the following formula, where a positive adjustment is in the clockwise direction. : \Delta Q = - \frac, where *''n'' is 1.85 for Hazen-Williams and *''n'' is 2 for Darcy–Weisbach. The clockwise specifier (c) means only the flows that are moving clockwise in our loop, while the counter-clockwise specifier (cc) is only the flows that are moving counter-clockwise. This adjustment doesn't solve the problem, since most networks have several loops. It is okay to use this adjustment, however, because the flow changes won't alter condition 1, and therefore, the other loops still satisfy condition 1. However, we should use the results from the first loop before we progress to other loops. An adaptation of this method is needed to account for water reservoirs attached to the network, which are joined in pairs by the use of 'pseudo-loops' in the Hardy Cross scheme. This is discussed further on the Hardy Cross method site. The modern method is simply to create a set of conditions from the above Kirchhoff laws (junctions and head-loss criteria). Then, use a
Root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbe ...
to find ''Q'' values that satisfy all the equations. The literal friction loss equations use a term called ''Q''2, but we want to preserve any changes in direction. Create a separate equation for each loop where the head losses are added up, but instead of squaring ''Q'', use , ''Q'', ·''Q'' instead (with , ''Q'', the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of ''Q'') for the formulation so that any sign changes reflect appropriately in the resulting head-loss calculation.


Probabilistic network analysis

In many situations, especially for real water distribution networks in cities (which can extend between thousands to millions of nodes), the number of known variables (flow rates and/or head losses) required to obtain a deterministic solution will be very large. Many of these variables will not be known, or will involve considerable uncertainty in their specification. Furthermore, in many pipe networks, there may be considerable variability in the flows, which can be described by fluctuations about mean flow rates in each pipe. The above deterministic methods are unable to account for these uncertainties, whether due to lack of knowledge or flow variability. For these reasons, a probabilistic method for pipe network analysis has recently been developed, based on the maximum entropy method of Jaynes. In this method, a continuous relative entropy function is defined over the unknown parameters. This entropy is then maximized subject to the constraints on the system, including Kirchhoff's laws, pipe friction properties and any specified mean flow rates or head losses, to give a probabilistic statement (
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
) which describes the system. This can be used to calculate mean values (expectations) of the flow rates, head losses or any other variables of interest in the pipe network. This analysis has been extended using a reduced-parameter entropic formulation, which ensures consistency of the analysis regardless of the graphical representation of the network. A comparison of Bayesian and maximum entropy probabilistic formulations for the analysis of pipe flow networks has also been presented, showing that under certain assumptions (Gaussian priors), the two approaches lead to equivalent predictions of mean flow rates. Other methods of
stochastic optimization Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective functi ...
of water distribution systems rely on
metaheuristic In computer science and mathematical optimization, a metaheuristic is a higher-level procedure or heuristic designed to find, generate, or select a heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimizat ...
algorithms, such as
simulated annealing Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It ...
and
genetic algorithms In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gen ...
.


See also


References


Further reading

*N. Hwang, R. Houghtalen, "Fundamentals of Hydraulic Engineering Systems" Prentice Hall, Upper Saddle River, NJ. 1996. *L.F. Moody, "Friction factors for pipe flow," Trans. ASME, vol. 66, 1944. *C. F. Colebrook, "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws," Jour. Ist. Civil Engrs., London (Feb. 1939). *Eusuff, Muzaffar M.; Lansey, Kevin E. (2003)
"Optimization of Water Distribution Network Design Using the Shuffled Frog Leaping Algorithm"
''Journal of Water Resources Planning and Management''. 129 (3): 210-225. {{Hydraulics Fluid dynamics Hydraulics Hydraulic engineering Networks Piping