Hardy Cross method
The Hardy Cross method is an application of continuity of flow and continuity of potential to iteratively solve for flows in a pipe network.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.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.
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:
hf=kQn
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 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
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 ⋅ I
is equivalent to
h f = k ⋅ 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.