Hardy Cross method

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.

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.

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.

[4] 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, is equivalent to 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.

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] 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.

[5] If the initial guess of flow rates in each pipe is correct, the change in head over a loop in the system,

The Hardy Cross method is useful because it relies on only simple mathematics, circumventing the need to solve a system of equations.

The Hardy Cross method iteratively corrects for the mistakes in the initial guess used to solve the problem.

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.

The Hardy Cross method can be used to calculate the flow distribution in a pipe network.

We will consider n to be 2, and the head loss per unit flow r, and initial flow guess for each pipe as follows: We solve the network by method of balancing heads, following the steps outlined in method process above.

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

It is found to be 60 in both loops (due to symmetry), as shown in the figure.

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.

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

The total head loss in loop 2-3-4 will also be balanced (again due to symmetry).

In this case, the method found the correct solution in one iteration.