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The Hazen–Williams equation is an empirical relationship that relates the flow of water in a pipe with the physical properties of the pipe and the pressure drop caused by friction. It is used in the design of water pipe systems [1] such as fire sprinkler systems, [2] water supply networks, and irrigation systems.
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:
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) which describes the system. This can be used to calculate mean values (expectations) of the flow rates ...
The flow rate can be converted to a mean flow velocity V by dividing by the wetted area of the flow (which equals the cross-sectional area of the pipe if the pipe is full of fluid). Pressure has dimensions of energy per unit volume, therefore the pressure drop between two points must be proportional to the dynamic pressure q.
Because hydraulic calculations for gridded systems require an iterative process to balance the water flow through all possible water paths, these calculations are most often performed by computer software. In practice, most calculations on all types of piping networks are performed by computer software.
In wide rectangular channels, the hydraulic radius is approximated by the flow depth. The hydraulic radius is not half the hydraulic diameter as the name may suggest, but one quarter in the case of a full pipe. It is a function of the shape of the pipe, channel, or river in which the water is flowing.
The flow coefficient of a device is a relative measure of its efficiency at allowing fluid flow. It describes the relationship between the pressure drop across an orifice valve or other assembly and the corresponding flow rate. Mathematically the flow coefficient C v (or flow-capacity rating of valve) can be expressed as
The laminar flow through a pipe of uniform (circular) cross-section is known as Hagen–Poiseuille flow. The equations governing the Hagen–Poiseuille flow can be derived directly from the Navier–Stokes momentum equations in 3D cylindrical coordinates ( r , θ , x ) by making the following set of assumptions: