Pipe Flow Calculations 1: the Entrance Length for Fully Developed Flow

Equations for analyzing pipe flow, such as the Darcy Weisbach equation for frictional head loss, often apply only to the fully developed flow portion of the pipe flow. If the total pipe length is large compared to the entrance length, then the effect of the entrance length can usually be neglected and the total pipe length can be used in calculations. If the total pipe length is relatively short in comparison with the entrance length, however, then the entrance region may need to be analyzed separately. An estimate of the entrance length is sometimes needed in order to determine how to proceed with pipe flow calculations. The Reynolds number for pipe flow is needed to calculate the entrance length for turbulent flow or for laminar flow.

The entrance length for pipe flow is a function of Reynolds number for both turbulent flow and laminar flow. You probably recall that turbulent flow will occur for flow in a pipe or duct if the Reynolds Number (Re) is greater than 4000. Most pipe flow of gases and liquids with a viscosity similar to water is turbulent flow.

For turbulent flow the entrance length, L_e, can be estimated from the equation: L_e/D = 4.4Re^1/6.

where Re = DVρ/μ for flow in circular pipes and Re = 4R_HVρ/μ for flow in non-circular ducts, and

D = pipe diameter, ft

V = average flow velocity in pipe (= Q/A), ft/sec

ρ = fluid density, slugs/ft³

μ = fluid viscosity, lb-sec/ft²

R_H = hydraulic radius, ft, where R_H = A/P, and

A = cross-sectional area normal to flow, ft²

P = wetted perimeter of pipe or duct cross-section, ft

Laminar flow in pipes and ducts takes place for Reynolds Number less than 2100, with Reynolds Number as defined just above. Laminar flow in pipes will occur only for very viscous fluids and/or very slow flows.

For Laminar Flow, the entrance length, L_e, can be estimated from the equation: L_e/D = 0.06 Re.

Consider flow of 1.2 cfs of water at 50^o F through a 4" diameter pipe. What would the entrance length be for this flow?

Solution: The density and viscosity of water at 50^oF are: ρ = 1.94 slugs/ft³ and μ = 2.34 x 10^-5.

The velocity, V, can be calculated from V = Q/A = 1.2/(πD²/4) = 1.2/[π(1/3)²/4] = 13.4 ft/sec.

Substituting values into Re = DVρ/μ, gives: Re = (1/3)(13.4)(1.94)/(2.34 x 10^-5) = 3.79 x 10⁵.

The value of Reynolds number is greater than 4000, so this is turbulent flow, and the entrance length can be estimated from the equation: L_e/D = 4.4 Re^1/6.

Thus: L_e/(1/3) = 4.4[(3.79 x 10⁴)^1/6], Le/(1/3) = 37.4, and L_e = 12.5 ft.