Case 52: Water - Round-edged Orifice Loss Coefficient in a Straight Pipe

Reference: Pipe Flow – A Practical and Comprehensive Guide, 2012, Publisher Wiley, Donald C. Rennels, Hobart M. Hudson, Chapter 13, Page 142

Pipe Flow Expert File: Case_52_Water_Round_Edged_Orifice_Straight_Pipe.pfe

Problem Description:

A straight pipe contains a round-edged single-hole orifice. Find the k0 loss coefficient for the orifice with a rounding radius specified as a ratio against the diameter of the orifice (r/d0).

Diagram??Description automatically generated

Use different orifice diameter to pipe diameter ratios ( = d0/d), with varying degrees of rounding to compare the calculated loss coefficient (k0).

The published data uses different calculation methods for comparison:

Donald C. Rennels, Hobart M. Hudson Equation, Equation 13.6, ASME Fluid Meters, and Alvi et al.

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Pipe Flow Expert Parameters:

Fluid Data: Water at, 68 °F

Pipe Data: Internal Diameter 4 inches, wall thickness 0.237 inches, roughness 1881 micro-inches

The following 45 systems with an inflow demand of 0.4 ft3/sec were used to model orifices with orifice / pipe diameter ratios ( = d0/d1) from 0.1 through to 0.9.

System #	r/do (Rounding Radius / Orifice Diameter)
1 – 9	1.00
10 – 18	0.25
19 – 27	0.18
28 – 36	0.125
37 – 45	0.08

Result Comparison:

Pipe Flow Expert Calculated Results and Published Graph Readings of Orifice Loss Coefficient (k0):

Orifice Diameter / Pipe Diameter (d0/d1)	Rounding Radius / Orifice Diameter (r/d0)	Donald C. Rennels, Hobart M. Hudson (k0)	ASME Fluid Meters Low Flow Nozzle (k0)	ASME Fluid Meters High Flow Nozzle (k0)	Alvi et al. (k0)	Pipe Flow Expert (k0)
0.1	1.0	1.01	-	-	-	1.01
0.2	1.0	0.95	0.95	-	-	0.95
0.3	1.0	0.85	0.86	-	-	0.85
0.4	1.0	0.72	0.74	-	-	Rounding Limit Exceeded
0.45	1.0	-	-	0.68	-	-
0.5	1.0	0.58	0.61	-	-	Rounding Limit Exceeded
0.6	1.0	0.42	-	0.47	-	Rounding Limit Exceeded
0.7	1.0	0.27	-	0.33	-	Rounding Limit Exceeded
0.8	1.0	0.13	-	0.19	-	Rounding Limit Exceeded
0.9	1.0	0.04	-	-	-	Rounding Limit Exceeded
0.1	0.25	1.32	-	-	-	1.32
0.2	0.25	1.25	-	-	1.255	1.25
0.3	0.25	1.14	-	-	-	1.14
0.4	0.25	0.99	-	-	1.06	0.99
0.5	0.25	0.81	-	-	-	0.81
0.6	0.25	0.62	-	-	0.625	0.61
0.7	0.25	0.41	-	-	-	Rounding Limit Exceeded
0.8	0.25	0.22	-	-	0.28	Rounding Limit Exceeded
0.9	0.25	0.07	-	-	-	Rounding Limit Exceeded
0.1	0.18	1.48	-	-	-	1.48
0.2	0.18	1.41	-	-	1.37	1.41
0.3	0.18	1.29	-	-	-	1.29
0.4	0.18	1.12	-	-	1.12	1.12
0.5	0.18	0.93	-	-	-	0.93
0.6	0.18	0.71	-	-	0.695	0.71
0.7	0.18	0.48	-	-	-	0.48
0.8	0.18	0.26	-	-	0.31	Rounding Limit Exceeded
0.9	0.18	0.09	-	-	-	Rounding Limit Exceeded
0.1	0.125	1.67	-	-	-	1.67
0.2	0.125	1.59	-	-	1.57	1.59
0.3	0.125	1.45	-	-	-	1.45
0.4	0.125	1.28	-	-	1.275	1.28
0.5	0.125	1.06	-	-	-	1.06
0.6	0.125	0.82	-	-	.79	0.82
0.7	0.125	0.56	-	-	-	0.56
0.8	0.125	0.31	-	-	-	0.31
0.9	0.125	0.10	-	-	-	Rounding Limit Exceeded
0.1	0.08	1.88	-	-	-	1.88
0.2	0.08	1.79	-	-	1.795	1.79
0.3	0.08	1.65	-	-	-	1.65
0.4	0.08	1.46	-	-	1.465	1.46
0.5	0.08	1.22	-	-	-	1.22
0.6	0.08	0.95	-	-	-	0.95
0.7	0.08	0.66	-	-	-	0.66
0.8	0.08	0.37	-	-	-	0.37
0.9	0.08	0.12	-	-	-	Rounding Limit Exceeded

Graphical Comparison of Results:

Commentary:

The published k0 loss coefficients compare well with the calculated results.

Rounding Limit: There is a point where circular rounding is limited by the available radial distance between the pipe wall and the face of the orifice.

The dashed red line on the above graph plots the rounding limit cut off point across each r/do line. Points below the dashed red line imply elliptical rounding and calculations for this are not supported.

The Pipe Flow Expert software reports a ‘Rounding Limit Exceeded’ message for any defined orifice data that is in this region.

Note: Head Loss in m fluid = (k0 * v2) / 2g

where v = fluid velocity in m/s at the entrance to the orifice, g = acceleration due to gravity in m/s2
k0 is not the same as a standard k value (which is used in formulas where v = velocity in the pipe)