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Reynolds Number
The Reynolds Number is used as an input and constraint for different design programs. This characterizes the flow and determines what equations can be used for different elements of the plant design.
The Reynolds Number can be calculated based on different known variables. The pipe transition used for plant design is 2100, which is the point flow goes from laminar to turbulent. The different equations were calculated based on the fundamental equation, as follows:

Friction Coefficient
The friction coefficient is used to determine losses associated with the tube. Two equations are used depending on whether the fluid is turbulent or laminar (greater than the transition flow or less than the transition flow).
For turbulent flow:

For laminar flow:

Head Loss
Head loss is calculated for major losses and minor losses.
Major losses are calculated based on the flow rate and the pipe dimensions. The major losses are caused primarily by shear in the pipes.

Minor losses are caused by flow expansions in the system components, such as valves, tees, and bends, are calculated similarly to major losses, except length is not a factor and a minor loss coefficient is used. The minor loss coefficient is coefficients for various system components are entered in the Design Assumptions and Minor Loss Coefficients programs.

Total head loss is used in some equations. It is simply the sum of major and minor losses, and is denoted as h.l.
Head loss is also calculated independent of the velocity head. This is used in the Entrance Tank program.

The total head loss in the manifold is used in the Exit Channel, Launder, and Sludge Drain program.

Orifice Equations
Orifice equations are used in the Entrance Tank, Exit Channel, Flow Control Module, and Launder. These equations include simple calculations for the area of a circle (A.circle) and the diameter of a circle (D.circle) given an area.
The flow rate through the orifice is calculated using a Pi value found in the Design Assumptions.

This equation is rearranged to calculate the area of the orifice (A.orifice) and for h (h.orifice).
The number of orifices can be calculated by dividing the area of the orifice by the area of a circle given the diameter of the orifice.
The diameter of the orifice can also be found the pipe characteristics.

Flow Equations
This section of the Fluids Functions calculates the flow rate based on different parameters.
The flow rate can be determined based on the Hagen-Poiseuille equation, which includes head loss.

The flow rate can also be calculated if a pipe roughness is given.

Flow through a pipe under either laminar or turbulent flow when there are only major losses is also calculated. This flow is equal to Q.hagen if it is less than the flow rate of the transition flow, or Q.SJ if it is larger.
Finally, the flow rate can be calculated given minor head losses.

Pipe Equations
This section of the Fluids Functions calculates the diameter of the pipe needed given different known variables. These mostly include loops that give you the available pipe size when the calculated diameter is entered. The nominal pipe size is also returned when a pipe specification is given (ND.Manifold).
The diameter of the orifice (D.Orifices) can also be calculated given head water and launder specs by using the D.circle equation as a function of A.orifice.

Weir Headloss
The head loss from a weir is calculated given a flow rate and pipe diameter.

 
This formula calculates the area of a port with given energy dissipation rate to ensure flocs will not break up. The area of the floc port is calculated give given the flow of the plant, the size of the vena contracta at the orifice, and specific energy dissipation rate.

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$$
A_{Port} = {1 \over {Pi_{VenaContractaOrifice} }} * ({{K_{Exit}  * Q^3 } \over {2 * \pi _{FlocCell}  * ED_{L\arg eFloc} }})^{{2 \over 7}} 
$$