## Equivalent Velocity Head Factors for Pipe Fittings

The 2nd edition of the CCPS Guidelines for Pressure Relief and Effluent Handling^{1} recommends the work of Darby^{2} for use in evaluating relief piping systems. We have already touched on the recommendations regarding roughness factors. Here we want to look at the equivalent velocity head factor for fittings.

**K-factor correlations.** Darby^{2} actually provides a good summary of the different approaches to determining friction losses in valves and fittings:

There are several “correlation” expressions for K_{f}… The “3-K” method is recommended, because it accounts directly for the effect of both Reynolds number and fitting size on the loss coefficient and more accurately reflects the scale effect of fitting size than the [Hooper] 2-K method. For highly turbulent flow, the Crane method agrees well with the 3-K method but is less accurate at low Reynolds numbers and is not recommended for laminar flow. The [constant] loss coefficient and [equivalent length] (L/D)_{eq} methods are more approximate but give acceptable results at high Reynolds numbers and when losses in valves and fittings are “minor losses” compared to the pipe friction. They are also appropriate for first estimates in problems that require iterative solutions.

Just as examples, constant loss coefficients for fittings are published by Perry’s^{3} and in API Standard 521^{4} (although there is a note in API Std 521 Table 9: “K can vary with nominal pipe diameter. The values above are typical only.”)

In relief system analysis, laminar or transition flows are rare, and so the methods commonly employed are not focused on accuracy at low Reynolds numbers. On the other hand, the scaling effects are important. Crane notes “The resistance coefficient K would theoretically be a constant for all sizes of a given design or line of valves and fittings if all sizes were geometrically similar. However, geometric similarity is seldom, if ever, achieved because the design of valves and fittings is dictated by manufacturing economies, standards, structural strength, and other considerations.”^{5}

Because of the wider applicability to the piping systems we encounter (particularly for the scaling effects), we have employed the Hooper 2-K method^{6} for determining equivalent velocity head factors for fittings. In the 2-K method, there are two coefficients, one related to the Reynolds number and one related to the scaling factor, that determine the equivalent velocity head factor. Hooper indicates “The two-K method takes these dependencies [of Reynolds number and of the exact geometry of the fitting] into account in the following equation:”

K_{1} = K for the fitting at N_{Re} = 1 [dimensionless],

K_{∞}= K for a large fitting at N_{Re} = ∞ [dimensionless], and

D = Internal diameter of the attached pipe [in]

One can see that the (1+1/D) term is the scaling factor for the size of the fitting. Darby’s improvement on this scaling factor is to introduce another coefficient for the scaling, and to make the scaling exponential with respect to the nominal pipe size (NPS):

Darby indicates with respect to the table of 3-K Constants for Loss Coefficients for Valves and Fittings, “The values of K_{1} are mostly those of the Hooper 2-K method, and the values of K_{∞} were mostly determined from the Crane data. However, since there is no single comprehensive data set for many fittings over a wide range of sizes and Reynolds numbers, some estimation was necessary for some values. Note that the values of K_{d} are all very close to 4.0, and this can be used to scale known values of K_{f} for a given pipe size to apply to other sizes.”

**Comparison of Results.** Before updating our calculation methods, we performed a comparison between the different methods for various pipe sizes we commonly encounter to see what the net effect of this change would mean. Below are the comparisons for four common fittings: 90° elbows with a long radius (r/d=1.5), 45° standard threaded elbows, and full bore fully opened (β=1) gate and globe valves. For a common comparison basis, we have used standard schedule piping, fully turbulent flow (Re=10^{7}), new commercial steel piping roughness (ε=0.0018 in), and a friction factor determined by the Colebrook equation:

Figure – K for 90° Elbow – long radius

*Note: Darby publishes coefficients for r/d=1 and r/d=2, but not r/d=1.5; therefore, an interpolation based on the turbulent flow coefficients published by Crane was used. The resultant 3-K coefficients used here are K _{1}=800, K_{∞}=0.06475, and K_{d}=3.935*

Figure – K for 45° Elbow – standard, threaded

Figure – K for Gate Valve – full bore (β=1), fully opened

Figure – K for Globe Valve – full port (β=1), fully opened

As one can see from the graphs, the differences in the equivalent velocity head factors are minor for the common pipe sizes of 1 – 10” NPS. Even the differences outside of that range could be considered relatively minor given the general accuracy of pipe flow calculations in the first place.

**Significant Error for Hooper 2K Flanged/Welded Run-through Tee.** There was one fitting where we found significant differences, and this was for a tee with flow through the body and no flow through the branch (“run-through”). In Crane, the picture of the fitting being represented is a standard threaded tee. In Darby, there are separate values for “threaded”, “flanged”, and a third description for “stub-in”, reinforcing that the first two tees are for the same (or similar) inlet/outlet/branch sizes. In Hooper, there are separate values for “screwed”, “flanged or welded”, and “stub-in”. The stub-in values are as expected (that is, no real resistance except at low Reynolds Numbers) and similar; however, the threaded and flanged fittings were found to be unexpectedly different.

Figure – K for Run-Through Tees

It is clear there is general agreement between Darby and Crane for the standard, threaded tee. At first glance, the values for the Hooper fittings appear to be incorrectly labeled; however, we confirm that the run-through flanged tees are predicted by Hooper’s 2-K method to have much higher velocity head factors than the threaded tees. It seems likely that the coefficients for flanged or welded run-through tees were incorrectly reported in the original Hooper 1981 article.

There is quite a bit of research on the energy losses in tees, particularly for combining or dividing flows. The run-through flow is a limiting condition in these tests; therefore, it should provide an independent check on the order of magnitude expected. Our focus was on validating flanged or welded tees as we encounter these most often. Abou-Haidar and Dixon^{8} present experimental data for equivalent velocity head factors for welded pipes having sharp edges at multiple angles over a range of flows. Their Figure 6 provides K values for 90° branches, with q=0 representing no flow from the branch and K_{23} being the main flow through the tee, shown as a value of approximately 0.05:

Their Figure 13 provides comparisons with other researchers results, again q=0 representing no flow from the branch and K_{23 } being the main flow through the tee, shown as a value of approximately 0.05:

Based on these data, we have confidence in the K values for tees estimated by Darby’s 3-K method coefficients.

**Conclusion.** In conclusion, the Darby 3-K method produces equivalent velocity head factors for common fittings in common pipe sizes that are similar to those produced by the Crane method and the Hooper 2-K method. There is a significant overestimation of the equivalent velocity head factor for run-through flanged/welded tees in the 1981 article by Hooper, which we think may be a transcription error in the article. Overall, based on the comparisons we have performed, we see no significant changes in employing the Darby 3-K method as recommended by the CCPS Guidelines for Pressure Relief and Effluent Handling (2nd edition); nonetheless, we will be doing so to keep our calculations up-to-date and to employ better estimates for tees.

It is important to note that the discussion above was in regard to fittings having a (relatively) constant diameter. For fittings with changes in diameter, Darby references the work of Hooper^{7}.

Also, as an aside, we looked at the effects of using the actual inner diameter versus the nominal pipe size in the 3-K method, and found minor differences, well within ±2% for common pipe sizes. This was expected as there is not a significant difference between the nominal pipe size and the actual inner diameter, and the parameter is raised to the 0.3 power.

[1] AIChE Center for Chemical Process Safety. “CCPS Guidelines for Pressure Relief and Effluent Handling Systems”. 2nd Edition, 2017.

[2] Darby R.

*Chemical Engineering Fluid Mechanics*. 2001; Boca Raton: CRC Press. pp. 207-213.

[3] Tilton JN. “Fluid and Particle Dynamics”. In Perry RH and Green DW.

*Perry’s Chemical Engineers’ Handbook*(pp. 6.1-6.54). 1997; New York: McGraw Hill.

[4] American Petroleum Institute. “API Standard 521: Pressure-relieving and Depressuring Systems”. 6th Edition, 2014 Jan.

[5] Crane.

*Technical Paper 410: Flow of Fluids*. 2009; Joliet: Crane Company.

[6] Hooper BW. “The Two-K Method Predicts Head Losses in Pipe Fittings.”

*Chemical Engineering*. (August 24, 1981) pp.96-100.

[7] Hooper BW. “Calculate Head Loss Caused by Change in Pipe Size.”

*Chemical Engineering*. (November 1988) pp.89-92.

[8] Abou-Haidar NI, Dixon SL. “Pressure Losses in Combining Subsonic Flows Through Branched Ducts”.

*Transactions of the ASME, 90-GT-134*; Presented at the Gas Turbine and Aeroengine Congress and Exposition, June 11-14, 1990, Brussels, Belgium.

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