Pressure Drop in Pipes - Detailed Discussion

Depending on the viscosity model and pipe friction model selected, the calculation of pressure drop in a pipe differs.

Darcy-Weisbach Loss Model

AFT Fathom utilizes the Darcy-Weisbach loss model to relate the Darcy Friction Factor, the pipe geometry, fluid density, and fluid velocity to pressure drop in the pipe. The Darcy Friction Factor differs from the Fanning Friction Factor by a factor of 4.

(1)

This model requires the calculation of a friction factor, as described below.

Roughness-Based Methods - This method calculates the friction factor based on the roughness of the pipe wall. Different equations are used based on flow regime - laminar flow uses the standard laminar relationship, while turbulent flow uses the implicit Colebrook-White equation. In the transition range between laminar and turbulent flow, a linearly interpolated value is used. The default transition Reynold's Numbers can be modified in the Environmental Properties panel.

(2)

Absolute Roughness (default) - The absolute average roughness height ε is specified directly.
Relative Roughness - The roughness is specified as a ratio ε/D.
Hydraulically Smooth - The ratio ε/D is set equal to zero.

Explicit Friction Factor - The friction factor to be used in the Darcy-Weisbach equation is specified directly.

Frictionless

The pipe will not have any pressure drop across it. This is inherently unrealistic behavior, but can be useful for troubleshooting purposes.

Resistance

The pipe resistance relates head loss to volumetric flow rate. In equation form, the head loss is:

(3)

Using standard relationships, the friction pressure drop is related to volumetric and mass flow rate as follows:

(4)

Non-Newtonian Pressure Loss

The pressure loss behavior for Non-Newtonian fluids depends on the specific viscosity model selected in Analysis Setup. See Non-Newtonian fluids for more information.

Helical Tubes

For helical tubes the pressure drop is modified from the equivalent "straight pipe" pressure drop via the following relationship (Ito 1959Ito, H., "Friction factors for pipe flow," Journal of Basic Engineering, Vol. 81, pp. 123-126, 1959.):

(5)

Hazen-Williams Method

AFT Fathom also offers the Hazen-Williams method of specifying irrecoverable loss information. The traditional Hazen-Williams formula (Zipparro and Hasen editors, 1993Zipparro, V. J., and H, Hasen, editors, Davis’ Handbook of Applied Hydraulics, Fourth Edition, McGraw-Hill. New York, NY, 1993. , page 2.7)

(6)

where

V = Velocity (ft/s)
C_HW = Hazen-Williams factor
R_h = Hydraulic Radius (cross-sectional area divided by wetted perimeter)
S = Hydraulic Gradient (ΔH/L)

can be modified to a more convenient head loss form for AFT Fathom:

(7)

where

V = Velocity (ft/s)
C_HW = Hazen-Williams factor
D = Inner Diameter in feet (four times R_h for circular pipe)
L = Length of Pipe (feet)

Also see Hazen Williams NFPA

MIT Equation for Crude Oil

The MIT Equation is appropriate for crude oil and is given by the following equation (Pipe Line Rules of Thumb HandbookPipe Line Rules of Thumb Handbook, Gulf Publishing, Houston, TX.):

(8)

(9)

(10)

where:

dP = pressure drop (psi)
L = length (miles)
f = friction factor, MIT
Q = volumetric flow rate (barrels/day)
s = specific gravity
d = inside diameter (inches)
ν = kinematic viscosity (centistokes)

The friction factor calculated by the MIT equations is not the same as the Darcy friction factor. Once the pressure drop (dP) in the pipe is determined, AFT Fathom uses the Darcy equation to back calculate the equivalent Darcy friction factor.

The MIT equation defines the lower bound for the laminar equation as r=0.1, but offers no explanation as to the validity of calculations below this point. AFT Fathom continues to use the laminar flow equation for Reynolds numbers below 0.1. The older version of the MIT equations did not have this lower limit, and the results from AFT Fathom 11 agree closely with values from the older equations at these lower Reynolds number values.

For Reynolds numbers in the Indeterminate flow regime, the value for the MIT friction factor is determined by linearly interpolating between the friction factors for the highest laminar Reynolds number and the lowest turbulent Reynolds number.

Miller Turbulent Method

The Miller Turbulent method is appropriate for light hydrocarbons and is given by the following equation (Pipe Line Rules of Thumb HandbookPipe Line Rules of Thumb Handbook, Gulf Publishing, Houston, TX.):

(11)

where:

dP = pressure drop (psi)
L = length (miles)
Q = volumetric flow rate (barrels/hour)
ρ = density (lbm/ft3)
ρ_water = density of water (62.3 lb_m/ft3)
d = diameter (inches)
μ = dynamic viscosity (centipoise)

Design Factors

In each pipe you can specify a Design Factor for the pipe friction. This is a multiplier that is applied to the friction factor calculated with the preceding methods.