Henry Darcy and His Law

The Darcy-Weisbach Equation

Glenn Brown
Oklahoma State University
6/22/00

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The Darcy-Weisbach equation is now considered the best empirical relation for pipe-flow resistance. In terms of head units it is,

(pipe friction)

where, hl is the head loss, f is the friction factor, L is the pipe length, V is the average flow velocity, and g is the acceleration of gravity.

In terms of pressure drop, Dp it is,

where r is the fluid density. The Darcy-Weisbach f is a complex function of the Reynolds Number and relative roughness. The Reynolds number, Re is defined as,

where m is the fluid absolute viscosity, and D is the pipe diameter. The relative pipe roughness is the ratio of the pipe surface roughness, e to its diameter, D, or e/D.

For laminar flow where Re < 2,000, pipe roughness is not a factor and,

f = 64/Re

For hydraulically smooth pipes (e = 0) such as glass, copper and plastic tubing in turbulent flow, use Blasius equation for f

(4,000 < Re < 100,000)

For rough pipe in turbulent flow you must use the Moody diagram to obtain f. That may require an iterative solution where a flow rate is guessed, f estimated and than a new flow calcuated.

An easier, and almost as accurate procedure as the Moody Diagram is to use the empirical formulas of Swamee and Jain, (J. of Hydraulics Division,. Proc. ASCE, pp 657-664, May 1976).

 

(10^-6 < e/D < 0.01; 5,000 < Re < 3x10^8)

(Note base 10 log used)

(3000<Re<3x10^8 ; 10^-6<e/d<.01)

(Re > 2,000)

(10-6 < e/D < 0.01; 5,000 < Re < 3x10^8)

where n is the kinematic viscosity, or m/r. The equation for f is a form of the Colebrook-White equation. The equation for Q is as accurate as the Moody diagram, while equations for hl and D are within 2%.

Equivalent Diameter for Noncircular Ducts: For noncircular ducts the hydraluic diameter, Dh is used as the characteristic length in Re and e/D.

Dh = 4 x area of flow / perimeter of duct in contact with fluid