Old Doesn't Mean Outdated

April 4, 2012
Some equations have stood the test of time.

Hundreds of attempts have been made to predict two-phase fluid flow. Yet more correlations continue to emerge because current ones don't work that well. Other important areas that seem to attract unending work to improve calculations include tray efficiency and critical heat flux in heat transfer.


In contrast, the long-established Darcy equation for pressure drop works well across the entire range of fluid flow from very low to very high velocities. The general form of the equation relates pressure drop, pipe geometry, velocity and fluid properties:

hL = ρfLv2/D2g

where hL is pressure drop in feet of flowing fluid, ρ is fluid density in lb/ft3, f is friction factor, L is length of pipe in ft, v is velocity in ft/sec, D is pipe diameter in ft, and g is the acceleration of gravity (32.2 ft/sec2).

However, the Darcy equation can pose pitfalls to the unwary.

First, you must understand the assumptions built into the equation, including:
• no phase change (i.e., vaporization);
• single-phase flow;
• insignificant variation in fluid properties across the flow system (constant density); and
• constant velocity (i.e., no changes in pipe diameter).

In addition, you must know the friction factor — determining f raises its own set of issues.

Crane Technical Paper No. 410, "Flow of Fluids Through Valves, Fittings, and Pipe," which first came out in 1942, remains one of the most useful sources of information on fluid flow evaluation. I wore out my original copy of the publication; the current version sitting on my desk is the 25th printing from 1991. Every engineer involved with basic fluid flow needs a copy of Crane 410 and needs to study it.

In real fluid-flow systems, two aspects that often generate the most grief are determining f and dealing with fittings.

Friction factor correlations for industrial straight pipe involve stream turbulence (expressed as a Reynolds number, NRE) and diameter. Crane 410 includes f tables and various forms of the Darcy equation for different units.

Handling a fitting involves generating an equivalent length of straight pipe that gives the same friction as the fitting. Here, complexity that can trip the unwary creeps into the analysis.

Due to lack of geometrical similarity in families of pipe fittings, friction factors are correlated against the equivalent length and pipe size (L/D). The idea is that a given fitting as a function of its size and shape has a resistance coefficient, K, independent of the flow rate, i.e.:

K = f (L/D)

Because K never changes for a fitting, the L/D ratio and, hence, equivalent length of the fitting varies inversely with f. Friction factor changes with velocity. Once fully developed turbulent flow occurs, the equivalent length doesn't change. But the velocity required to establish such flow can be surprisingly high for many systems. For example, at the beginning of turbulent flow in a 2-in. commercial steel pipe (NRE = 4,000), f ≈ 0.039. At fully developed turbulent flow (NRE = 1,500,000), f = 0.019.

Too often, engineers use simple software that approaches calculations incorrectly. It asks for an equivalent length and plugs in that value, which applies to only one flow condition, at many different rates.

Crane 410 deals with this by having two sets of equations. Set 1 (Equation 3-5) gives the head loss for straight pipe. Set 2 (Equation 3-14) gives the head loss for fittings. You should separately calculate the head losses for the straight pipe and fittings and then sum them. You can add all the fittings in a system together to get a "total K" for the system and then perform a single calculation using Equation 3-14. Unfortunately, while this should be a relatively simple task for software to handle, it's often not done.

How important is this difference? The more fittings there are, the bigger the consequences. The tighter the system hydraulics, the more serious this issue is. My experience is that the most problems occur in gravity flow systems inside process plants. With gravity flow, head available often is limited and flow velocities frequently are low. Low velocities tend to emphasize the effect of errors in not updating equivalent lengths. Piping inside process units usually has many more fittings than runs to the tank farm.

Equivalent lengths and K values aren't the only sources of potential errors in using the Darcy equation. Next month, we'll look at the concept of velocity head, its impact on pressure drop and other uses of the concept.

ANDREW SLOLEY is a Chemical Processing contributing editor. You can e-mail him at [email protected]

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