Hardy Cross method

Last updated June 02, 2023

The Hardy Cross method is an iterative method for determining the flow in pipe network systems where the inputs and outputs are known, but the flow inside the network is unknown.^[1] The method was first published in November 1936 by its namesake, Hardy Cross, a structural engineering professor at the University of Illinois at Urbana–Champaign.^[2] The Hardy Cross method is an adaptation of the Moment distribution method, which was also developed by Hardy Cross as a way to determine the forces in statically indeterminate structures.

The introduction of the Hardy Cross method for analyzing pipe flow networks revolutionized municipal water supply design. Before the method was introduced, solving complex pipe systems for distribution was extremely difficult due to the nonlinear relationship between head loss and flow. The method was later made obsolete by computer solving algorithms employing the Newton–Raphson method or other numerical methods that eliminate the need to solve nonlinear systems of equations by hand.

History

In 1930, Hardy Cross published a paper called "Analysis of Continuous Frames by Distributing Fixed-End Moments" in which he described the moment distribution method, which would change the way engineers in the field performed structural analysis.^[3] The moment distribution method was used to determine the forces in statically indeterminate structures and allowed for engineers to safely design structures from the 1930s through the 1960s, until the development of computer oriented methods.^[3] In November 1936, Cross applied the same geometric method to solving pipe network flow distribution problems, and published a paper called "Analysis of flow in networks of conduits or conductors."^[1]

Derivation

The Hardy Cross method is an application of continuity of flow and continuity of potential to iteratively solve for flows in a pipe network.^[1] In the case of pipe flow, conservation of flow means that the flow in is equal to the flow out at each junction in the pipe. Conservation of potential means that the total directional head loss along any loop in the system is zero (assuming that a head loss counted against the flow is actually a head gain).

Hardy Cross developed two methods for solving flow networks. Each method starts by maintaining either continuity of flow or potential, and then iteratively solves for the other.

Assumptions

The Hardy Cross method assumes that the flow going in and out of the system is known and that the pipe length, diameter, roughness and other key characteristics are also known or can be assumed.^[1] The method also assumes that the relation between flow rate and head loss is known, but the method does not require any particular relation to be used.^[1]

In the case of water flow through pipes, a number of methods have been developed to determine the relationship between head loss and flow. The Hardy Cross method allows for any of these relationships to be used.

The general relationship between head loss and flow is:

h_{f}=k\cdot Q^{n}

where k is the head loss per unit flow and n is the flow exponent. In most design situations the values that make up k, such as pipe length, diameter, and roughness, are taken to be known or assumed and thus the value of k can be determined for each pipe in the network. The values that make up k and the value of n change depending on the relation used to determine head loss. However, all relations are compatible with the Hardy Cross method.^[4]

Head loss equation	Relation	k	n
Hazen-Williams equation	$h_{f}=L\cdot {\frac {10.67\quad Q^{1.85}}{C^{1.85}\quad d^{4.87}}}$	$L\cdot {\frac {10.67}{C^{1.85}\quad d^{4.87}}}$	1.85
Darcy-Weisbach equation	$h_{f}={\frac {8fLQ^{2}}{g\pi ^{2}d^{5}}}$	$L\cdot {\frac {8f}{g\pi ^{2}d^{5}}}$	2

It is also worth noting that the Hardy Cross method can be used to solve simple circuits and other flow like situations. In the case of simple circuits,

V=K\cdot I

is equivalent to

h_{f}=k\cdot Q^{n}

.

By setting the coefficient k to K, the flow rate Q to I and the exponent n to 1, the Hardy Cross method can be used to solve a simple circuit. However, because the relation between the voltage drop and current is linear, the Hardy Cross method is not necessary and the circuit can be solved using non-iterative methods.

Method of balancing heads

The method of balancing heads uses an initial guess that satisfies continuity of flow at each junction and then balances the flows until continuity of potential is also achieved over each loop in the system.^[1]

Proof (r denotes k)

The following proof is taken from Hardy Cross's paper, “Analysis of flow in networks of conduits or conductors.”,^[1] and can be verified by National Programme on Technology Enhanced Learning Water and Wastewater Engineering page,^[4] and Fundamentals of Hydraulic Engineering Systems by Robert J. Houghtalen.^[5]

If the initial guess of flow rates in each pipe is correct, the change in head over a loop in the system, $\Sigma rQ^{n}$ would be equal to zero. However, if the initial guess is not correct, then the change in head will be non-zero and a change in flow, $\Delta Q$ must be applied. The new flow rate, $Q=Q_{0}+\Delta Q$ is the sum of the old flow rate and some change in flow rate such that the change in head over the loop is zero. The sum of the change in head over the new loop will then be $\Sigma r(Q_{0}+\Delta Q)^{n}=0$ .

The value of $\Sigma r(Q_{0}+\Delta Q)^{n}$ can be approximated using the Taylor expansion.

\Sigma r(Q_{0}+\Delta Q)^{n}=\Sigma r(Q_{0}^{n}+nQ_{0}^{n-1}\Delta Q+...)=0

For a small $\Delta Q$ compared to $Q_{0}$ the additional terms vanish, leaving:

\Sigma r(Q_{0}^{n}+nQ_{0}^{n-1}\Delta Q)=0

And solving for $\Delta Q$

\Sigma rQ_{0}^{n}=-\Sigma nrQ_{0}^{n-1}\Delta Q

\Delta Q=-{\frac {\Sigma rQ_{0}^{n}}{\Sigma nrQ_{0}^{n-1}}}

The change in flow that will balance the head over the loop is approximated by $\Delta Q=-{\frac {\Sigma rQ_{0}^{n}}{\Sigma nrQ_{0}^{n-1}}}$ . However, this is only an approximation due to the terms that were ignored from the Taylor expansion. The change in head over the loop may not be zero, but it will be smaller than the initial guess. Multiple iterations of finding a new $\Delta Q$ will approximate to the correct solution.^[1]

Process

The method is as follows:

Guess the flows in each pipe, making sure that the total in flow is equal to the total out flow at each junction. (The guess doesn't have to be good, but a good guess will reduce the time it takes to find the solution.)
Determine each closed loop in the system.
For each loop, determine the clockwise head losses and counter-clockwise head losses. Head loss in each pipe is calculated using $h_{f}=rQ^{n}$ . Clockwise head losses are from flows in the clockwise direction and likewise for counter-clockwise.
Determine the total head loss in each loop, $\Sigma rQ^{n}$ , by subtracting the counter-clockwise head loss from the clockwise head loss.
For each loop, find $\Sigma nrQ^{n-1}$ without reference to direction (all values should be positive).
The change in flow is equal to ${\frac {\Sigma rQ^{n}}{\Sigma nrQ^{n-1}}}$ .
If the change in flow is positive, apply it to all pipes of the loop in the counter-clockwise direction. If the change in flow is negative, apply it to all pipes of the loop in the clockwise direction.
Continue from step 3 until the change in flow is within a satisfactory range.

Method of balancing flows (section incomplete)

The method of balancing flows uses an initial guess that satisfies continuity of potential over each loop and then balances the flows until continuity of flow is also achieved at each junction.

Advantages of the Hardy Cross method

Simple mathematics

The Hardy Cross method is useful because it relies on only simple mathematics, circumventing the need to solve a system of equations. Without the Hardy Cross methods, engineers would have to solve complex systems of equations with variable exponents that cannot easily be solved by hand.

Self correcting

The Hardy Cross method iteratively corrects for the mistakes in the initial guess used to solve the problem.^[1] Subsequent mistakes in calculation are also iteratively corrected. If the method is followed correctly, the proper flow in each pipe can still be found if small mathematical errors are consistently made in the process. As long as the last few iterations are done with attention to detail, the solution will still be correct. In fact, it is possible to intentionally leave off decimals in the early iterations of the method to run the calculations faster.

Example

The Hardy Cross method can be used to calculate the flow distribution in a pipe network. Consider the example of a simple pipe flow network shown at the right. For this example, the in and out flows will be 10 liters per second. We will consider n to be 2, and the head loss per unit flow r, and initial flow guess for each pipe as follows:

Pipe	Q12	Q13	Q23	Q24	Q34
r	1	5	1	5	1
Q guess (L/s)	5	5	0	5	5

We solve the network by method of balancing heads, following the steps outlined in method process above.

1. The initial guesses are set up so that continuity of flow is maintained at each junction in the network.

2. The loops of the system are identified as loop 1-2-3 and loop 2-3-4.

3. The head losses in each pipe are determined.

Loop 1-2-3	Q12	Q13	Q23
Head loss = $rQ^{2}$	25	125	0
Direction	Clockwise	Counter-clockwise	Clockwise

For loop 1-2-3, the sum of the clockwise head losses is 25 and the sum of the counter-clockwise head losses is 125.

Loop 2-3-4	Q23	Q24	Q34
Head loss = $rQ^{2}$	0	125	25
Direction	Counter-clockwise	Clockwise	Counter-clockwise

For loop 2-3-4, the sum of the clockwise head losses is 125 and the sum of the counter-clockwise head losses is 25.

4. The total clockwise head loss in loop 1-2-3 is $25-125=-100$ . The total clockwise head loss in loop 2-3-4 is $125-25=100$ .

5. The value of $\Sigma nrQ^{n-1}$ is determined for each loop. It is found to be 60 in both loops (due to symmetry), as shown in the figure.

6. The change in flow is found for each loop using the equation ${\frac {\Sigma rQ^{n}}{\Sigma nrQ^{n-1}}}$ . For loop 1-2-3, the change in flow is equal to $-100/60=-1.66$ and for loop 2-3-4 the change in flow is equal to $100/60=1.66$ .

7. The change in flow is applied across the loops. For loop 1-2-3, the change in flow is negative so its absolute value is applied in the clockwise direction. For loop 2-3-4, the change in flow is positive so its absolute value is applied in the counter-clockwise direction. For pipe 2-3, which is in both loops, the changes in flow are cumulative.

Pipe	Q12	Q13	Q23	Q24	Q34
Q (L/s)	6.66	3.33	3.33	3.33	6.66

The process then repeats from step 3 until the change in flow becomes sufficiently small or goes to zero.

3. The total head loss in Loop 1-2-3 is

Loop 1-2-3	Q12	Q13	Q23
Head loss = $rQ^{2}$	44.4	55.5	11.1
Direction	Clockwise	Counter-clockwise	Clockwise

Notice that the clockwise head loss is equal to the counter-clockwise head loss. This means that the flow in this loop is balanced and the flow rates are correct. The total head loss in loop 2-3-4 will also be balanced (again due to symmetry).

Loop 2-3-4	Q23	Q24	Q34
Head loss = $rQ^{2}$	11.1	55.5	44.4
Direction	Counter-clockwise	Clockwise	Counter-clockwise

In this case, the method found the correct solution in one iteration. For other networks, it may take multiple iterations until the flows in the pipes are correct or approximately correct.

Related Research Articles

In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n".

In mathematical physics and mathematics, the Pauli matrices are a set of three $2 \times 2$ complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs.

In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite-state machine (DFSM), or deterministic finite-state automaton (DFSA)—is a finite-state machine that accepts or rejects a given string of symbols, by running through a state sequence uniquely determined by the string. Deterministic refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943.

In power engineering, the power-flow study, or load-flow study, is a numerical analysis of the flow of electric power in an interconnected system. A power-flow study usually uses simplified notations such as a one-line diagram and per-unit system, and focuses on various aspects of AC power parameters, such as voltages, voltage angles, real power and reactive power. It analyzes the power systems in normal steady-state operation.

In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of all block codes in a unified way. Such limitations often take the form of bounds that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors.

Mehrotra's predictor–corrector method in optimization is a specific interior point method for linear programming. It was proposed in 1989 by Sanjay Mehrotra.

In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.

ALOPEX is a correlation based machine learning algorithm first proposed by Tzanakou and Harth in 1974.

The lattice Boltzmann methods (LBM), originated from the lattice gas automata (LGA) method (Hardy-Pomeau-Pazzis and Frisch-Hasslacher-Pomeau models), is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations directly, a fluid density on a lattice is simulated with streaming and collision (relaxation) processes. The method is versatile as the model fluid can straightforwardly be made to mimic common fluid behaviour like vapour/liquid coexistence, and so fluid systems such as liquid droplets can be simulated. Also, fluids in complex environments such as porous media can be straightforwardly simulated, whereas with complex boundaries other CFD methods can be hard to work with.

Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible.

In fluid dynamics, pipe network analysis is the analysis of the fluid flow through a hydraulics network, containing several or many interconnected branches. The aim is to determine the flow rates and pressure drops in the individual sections of the network. This is a common problem in hydraulic design.

In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are: (i) choose novel unconstrained coordinates, (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.

Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of -dimensional data items, a configuration of $points in -dimensional space is sought that minimizes the so-called stress function . Usually is or, i.e. the matrix lists points in or dimensional Euclidean space so that the result may be visualised. The function is a cost or loss function that measures the squared differences between ideal distances and actual distances in r -dimensional space. It is defined as:$

In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve parabolic and elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions. It is an example of an operator splitting method.

Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences. In economic theory, the non-linear least squares method is applied in (i) the probit regression, (ii) threshold regression, (iii) smooth regression, (iv) logistic link regression, (v) Box–Cox transformed regressors ( $).$

<span class="mw-page-title-main">Black–Scholes equation</span> Partial differential equation in mathematical finance

In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.

<span class="mw-page-title-main">Variance gamma process</span>

In the theory of stochastic processes, a part of the mathematical theory of probability, the variance gamma process (VG), also known as Laplace motion, is a Lévy process determined by a random time change. The process has finite moments distinguishing it from many Lévy processes. There is no diffusion component in the VG process and it is thus a pure jump process. The increments are independent and follow a variance-gamma distribution, which is a generalization of the Laplace distribution.

In machine learning, the vanishing gradient problem is encountered when training artificial neural networks with gradient-based learning methods and backpropagation. In such methods, during each iteration of training each of the neural networks weights receives an update proportional to the partial derivative of the error function with respect to the current weight. The problem is that in some cases, the gradient will be vanishingly small, effectively preventing the weight from changing its value. In the worst case, this may completely stop the neural network from further training. As one example of the problem cause, traditional activation functions such as the hyperbolic tangent function have gradients in the range $(0,1]$ , and backpropagation computes gradients by the chain rule. This has the effect of multiplying $n$ of these small numbers to compute gradients of the early layers in an $n$ -layer network, meaning that the gradient decreases exponentially with $n$ while the early layers train very slowly.

The Louvain method for community detection is a method to extract communities from large networks created by Blondel et al. from the University of Louvain. The method is a greedy optimization method that appears to run in time $where is the number of nodes in the network.$

References

1 2 3 4 5 6 7 8 9 Cross, H. (November 1936). "Analysis of flow in networks of conduits or conductors". Engineering Experiment Station. Bulletin No. 286.
↑ "Hardy Cross; Educator, analyst, engineer, philosopher". Archived from the original on August 9, 2011. Retrieved May 3, 2011.
1 2 Leonard K Eaton. "Hardy Cross and "The Moment Distribution Method"" . Retrieved April 10, 2011.
1 2 "Water & Wastewater Engineering". Archived from the original on March 12, 2008. Retrieved April 11, 2011.
↑ Robert J. Houghtalen (2009). Fundamentals of Hydraulic Engineering Systems. ISBN 9780136016380 . Retrieved April 10, 2011.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[HC-1] 1 2 3 4 5 6 7 8 9 Cross, H. (November 1936). "Analysis of flow in networks of conduits or conductors". Engineering Experiment Station. Bulletin No. 286.

[2] "Hardy Cross; Educator, analyst, engineer, philosopher". Archived from the original on August 9, 2011. Retrieved May 3, 2011.

[MDM-3] 1 2 Leonard K Eaton. "Hardy Cross and "The Moment Distribution Method"" . Retrieved April 10, 2011.

[IN-4] 1 2 "Water & Wastewater Engineering". Archived from the original on March 12, 2008. Retrieved April 11, 2011.

[FHES-5] Robert J. Houghtalen (2009). Fundamentals of Hydraulic Engineering Systems. ISBN 9780136016380 . Retrieved April 10, 2011.

[1]

[2]

[3]

[4]

[5]