Fully Developed Flow

Fully developed flow is when the viscous effects due to the shear stress between the fluid particles and pipe wall create a fully developed velocity profile for a fluid as it travels through the length of a straight pipe. The velocity of the fluid for a fully developed flow will be at its fastest at the center line of the pipe (equation 1 laminar flow), and the velocity of the fluid at the walls of the pipe will be at its slowest. Due to the change of velocity across the velocity profile it is common to describe the fluid velocity as an average velocity.

(Eq 1)  $v_C = \frac{2Q}{πR^2}$

vc = Maximum Velocity

Q = Flow Rate

R = Pipe Radius

As mentioned earlier the viscous effects are caused by the shear stress between a fluid and the wall of a pipe. This shear stress is always present despite how smooth the pipe is. Also, the shear stress between the fluid particles moving past one another is a product of the wall shear stress and the distance from the wall. Refer to equation 2 to calculate the shear stress between fluid particles for laminar flow.

(Eq 2)  $τ=\frac{2τ_wr}{D}$

τ = Shear Stress

τw = Shear Stress at the Wall

r = radial distance from the center of the pipe to point of interest

D = Pipe Diameter

Due to the shear stress on the fluid particles as the fluid moves past the pipe wall a pressure drop will occur as can be seen in equation 3.

(Eq 3)  $P_2=P_1-ΔP$

The viscous effects, pressure drop, and pipe length will affect the flow rate. To calculate the average flow rate, taking these into account, equation 4 would be used to calculate the flow rate for laminar flow.

(Eq 4)  $Q=\frac{πD^4ΔP}{128μL}$

L = Pipe Length

μ = Dynamic Viscosity

Entrance Length

When fluid first enters a pipe its flow is not fully developed. Instead the fluid has to travel a certain distance undisturbed before it becomes fully developed. This is also true when a fluid goes around a curve in the pipe system. The curve in the pipe will disrupt the velocity profile of the fluid, and it will need to travel a certain distance in a straight pipe to have a fully developed flow again. Refer to equation 5 to calculate the entrance length for laminar flow, and equation 6 to calculate the entrance length for turbulent flow.

(Eq 5)  $\frac{L}{D}=0.06·Re$

L = Entrance Length

D = Pipe Diameter

Re = Reynolds Number

(Eq 6)  $\frac{L}{D}=4.4(Re)^{1.6}$

Notice from the diagram that a boundary layer starts to form as the flow becomes developed. The boundary layer represents where the viscous effects are produced along the pipe wall to create the velocity profile. Also notice while the flow is developing there is a region where there is no viscous effects; this is called the inviscid core.

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