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Being able to determine and calculate primary fluid flow parameters such as pressure and velocity is essential to the successful design of any hydraulic network. The energy of an incompressible fluid can be expressed in an algebraic relation that relates its pressure, velocity and height. Employing the 1st law of thermodynamics allows us to set the energy at some initial point 1, to the energy at point 2. The result is the widely-used field equation shown below.

Eqn 1.

Where:

p = pressure

u = velocity

g = acceleration due to gravity

ρ = density of fluid

Because the flow of our water is internal, within a pipe, there exists a resistance at the walls of the pipe known as viscous shear that the flow must counter. The amount of energy the fluid exerts and essentially loses due to this frictional like effect is known as head loss. Head loss is expressed in a height which is typical in hydraulic applications because it gives the value in a number that is somewhat physical. Although seeming straightforward, head loss Is a difficult parameter to accurately calculate. Primarily, it is a function of the Reynolds number of the flow, velocity, inner pipe diameter, and the material of the pipe. The equation for head loss is given below.

Eqn 2.

Where:

h_{f} = head loss

L = length of pipe

D = inner diameter

f = friction factor

In our analysis, we will employ the use of the Moody friction factor because of its use in previous university courses. To determine the moody friction factor, one must first calculate the Reynolds number (Re) and therefore must know the velocity of the flow.Below is the relationship between flow rate (V) and velocity u.

Eqn 3.

Where:

V = flow rate

u = velocity

Once the velocity is known, the Reynolds number can be calculated as long as the type of fluid and its properties are known. The following equation allows for determination of the Reynold's number.

Eqn 4.

Where:

Re = Reynolds Number

μ = Dynamic Viscosity

The final piece of information needed to determine the friction factor is dependent on the material and surface of the pipe, the specific parameter is called the roughness factor. The roughness factor can directly be found from the moody diagram pertaining to various materials such as Cast Iron and Corrugated metal pipe.

Moody Diagram

Besides the head loss through what is assumed to be a straight pipe, there are losses associated with turning the fluid at junctions, fittings, valves, and entrance/exit points. For the sake of conciseness, these minor losses will be omitted and analyzed in greater detail in upcoming detailed reports where calculations are performed.