Case 57: Water – Round-edged Orifice Loss Coefficient Discharging to a Reservoir
Reference: Pipe Flow – A Practical and Comprehensive Guide, 2012, Publisher Wiley, Donald C. Rennels, Hobart M. Hudson, Chapter 12, Page 133
Pipe Flow Expert File: Case_57_Water_Round_Edged_Orifice_Discharge_To_Reservoir.pfe
Problem Description:
A straight pipe contains a round-edged single-hole orifice which discharges to a reservoir (atmosphere). The orifice diameter to pipe diameter is specified by the ratio d0/d1. The orifice rounding radius to orifice diameter is specified by the ratio r/d0.
Use different orifice rounding radius’ to orifice diameter ratios (r/d0) to compare the loss coefficient (k0) through the orifice with varying orifice diameter to pipe diameter ratios (d0/d).
Pipe Flow Expert Parameters:
Fluid Data: Water at, 68 °F
One hundred and seventy-nine individual systems with inflow demands of 0.01 ft3/sec were used to model a range of orifice rounding radius to orifice diameter ratios (r/d0) across a range of orifice diameter to pipe diameter ratios (d0/d).
Result Comparison:
Pipe Flow Expert Calculated Results and Published Graph Readings of Orifice Loss Coefficient (k0):
|
Orifice Diameter / Pipe Diameter (d0/d) |
Orifice Rounding Radius / Pipe Diameter (r/d0) |
Donald C. Rennels, Hobart M. Hudson (k0) |
Pipe Flow Expert (k0) |
|
0.1 |
1.0 |
1.027 |
1.027 |
|
0.2 |
1.0 |
1.024 |
1.024 |
|
0.3 |
1.0 |
1.021 |
1.0209 |
|
0.4 |
1.0 |
1.018 |
Rounding Limit Exceeded |
|
0.5 |
1.0 |
1.015 |
Rounding Limit Exceeded |
|
0.6 |
1.0 |
1.011 |
Rounding Limit Exceeded |
|
0.7 |
1.0 |
1.008 |
Rounding Limit Exceeded |
|
0.8 |
1.0 |
1.004 |
Rounding Limit Exceeded |
|
0.9 |
1.0 |
1.001 |
Rounding Limit Exceeded |
|
0.1 |
0.7 |
1.044 |
1.0442 |
|
0.2 |
0.7 |
1.041 |
1.0406 |
|
0.3 |
0.7 |
1.037 |
1.0369 |
|
0.4 |
0.7 |
1.033 |
1.0331 |
|
0.5 |
0.7 |
1.029 |
Rounding Limit Exceeded |
|
0.6 |
0.7 |
1.024 |
Rounding Limit Exceeded |
|
0.7 |
0.7 |
1.019 |
Rounding Limit Exceeded |
|
0.8 |
0.7 |
1.013 |
Rounding Limit Exceeded |
|
0.9 |
0.7 |
1.006 |
Rounding Limit Exceeded |
|
0.1 |
0.5 |
1.095 |
1.0946 |
|
0.2 |
0.5 |
1.091 |
1.0905 |
|
0.3 |
0.5 |
1.086 |
1.0862 |
|
0.4 |
0.5 |
1.081 |
1.0812 |
|
0.5 |
0.5 |
1.075 |
1.0751 |
|
0.6 |
0.5 |
1.067 |
Rounding Limit Exceeded |
|
0.7 |
0.5 |
1.057 |
Rounding Limit Exceeded |
|
0.8 |
0.5 |
1.043 |
Rounding Limit Exceeded |
|
0.9 |
0.5 |
1.024 |
Rounding Limit Exceeded |
|
0.1 |
0.3 |
1.269 |
1.2647 |
|
0.2 |
0.3 |
1.265 |
1.2594 |
|
0.3 |
0.3 |
1.259 |
1.2528 |
|
0.4 |
0.3 |
1.253 |
1.244 |
|
0.5 |
0.3 |
1.244 |
1.2315 |
|
0.6 |
0.3 |
1.232 |
1.2131 |
|
0.7 |
0.3 |
1.185 |
Rounding Limit Exceeded |
|
0.8 |
0.3 |
1.143 |
Rounding Limit Exceeded |
|
0.9 |
0.3 |
1.083 |
Rounding Limit Exceeded |
|
0.1 |
0.2 |
1.455 |
1.4552 |
|
0.2 |
0.2 |
1.449 |
1.4485 |
|
0.3 |
0.2 |
1.439 |
1.4393 |
|
0.4 |
0.2 |
1.426 |
1.426 |
|
0.5 |
0.2 |
1.406 |
1.4059 |
|
0.6 |
0.2 |
1.317 |
1.375 |
|
0.7 |
0.2 |
1.252 |
1.3263 |
|
0.8 |
0.2 |
1.252 |
Rounding Limit Exceeded |
|
0.9 |
0.2 |
1.145 |
Rounding Limit Exceeded |
|
0.1 |
0.15 |
1.601 |
1.6011 |
|
0.2 |
0.15 |
1.593 |
1.5934 |
|
0.3 |
0.15 |
1.582 |
1.5822 |
|
0.4 |
0.15 |
1.565 |
1.5654 |
|
0.5 |
0.15 |
1.539 |
1.5392 |
|
0.6 |
0.15 |
1.498 |
1.4984 |
|
0.7 |
0.15 |
1.433 |
1.4333 |
|
0.8 |
0.15 |
1.334 |
Rounding Limit Exceeded |
|
0.9 |
0.15 |
1.191 |
Rounding Limit Exceeded |
|
0.1 |
0.1 |
1.803 |
1.8027 |
|
0.2 |
0.1 |
1.794 |
1.7936 |
|
0.3 |
0.1 |
1.78 |
1.7795 |
|
0.4 |
0.1 |
1.758 |
1.7578 |
|
0.5 |
0.1 |
1.723 |
1.7231 |
|
0.6 |
0.1 |
1.668 |
1.6681 |
|
0.7 |
0.1 |
1.580 |
1.5800 |
|
0.8 |
0.1 |
1.446 |
1.4459 |
|
0.9 |
0.1 |
1.253 |
Rounding Limit Exceeded |
|
0.1 |
0.07 |
1.965 |
1.9648 |
|
0.2 |
0.07 |
1.955 |
1.9546 |
|
0.3 |
0.07 |
1.938 |
1.9383 |
|
0.4 |
0.07 |
1.913 |
1.9125 |
|
0.5 |
0.07 |
1.871 |
1.8708 |
|
0.6 |
0.07 |
1.804 |
1.8042 |
|
0.7 |
0.07 |
1.697 |
1.6971 |
|
0.8 |
0.07 |
1.534 |
1.5342 |
|
0.9 |
0.07 |
1.301 |
Rounding Limit Exceeded |
|
0.1 |
0.05 |
2.1 |
2.0999 |
|
0.2 |
0.05 |
2.089 |
2.0888 |
|
0.3 |
0.05 |
2.071 |
2.0707 |
|
0.4 |
0.05 |
2.042 |
2.0415 |
|
0.5 |
0.05 |
1.994 |
1.9939 |
|
0.6 |
0.05 |
1.917 |
1.9174 |
|
0.7 |
0.05 |
1.794 |
1.7942 |
|
0.8 |
0.05 |
1.607 |
1.6071 |
|
0.9 |
0.05 |
1.341 |
1.3408 |
|
0.1 |
0.03 |
2.2700 |
2.2699 |
|
0.2 |
0.03 |
2.2580 |
2.2578 |
|
0.3 |
0.03 |
2.2380 |
2.2378 |
|
0.4 |
0.03 |
2.2040 |
2.2040 |
|
0.5 |
0.03 |
2.1490 |
2.1488 |
|
0.6 |
0.03 |
2.0600 |
2.0597 |
|
0.7 |
0.03 |
1.9160 |
1.9160 |
|
0.8 |
0.03 |
1.6980 |
1.6980 |
|
0.9 |
0.03 |
1.3900 |
1.3896 |
|
0.1 |
0.02 |
2.377 |
2.377 |
|
0.2 |
0.02 |
2.364 |
2.3643 |
|
0.3 |
0.02 |
2.343 |
2.3425 |
|
0.4 |
0.02 |
2.306 |
2.3064 |
|
0.5 |
0.02 |
2.247 |
2.2465 |
|
0.6 |
0.02 |
2.149 |
2.1494 |
|
0.7 |
0.02 |
1.993 |
1.9925 |
|
0.8 |
0.02 |
1.755 |
1.755 |
|
0.9 |
0.02 |
1.42 |
1.4199 |
|
0.1 |
0.01 |
2.514 |
2.5136 |
|
0.2 |
0.01 |
2.5 |
2.5002 |
|
0.3 |
0.01 |
2.477 |
2.4767 |
|
0.4 |
0.01 |
2.437 |
2.4372 |
|
0.5 |
0.01 |
2.371 |
2.3712 |
|
0.6 |
0.01 |
2.264 |
2.2638 |
|
0.7 |
0.01 |
2.09 |
2.0901 |
|
0.8 |
0.01 |
1.827 |
1.8272 |
|
0.9 |
0.01 |
1.458 |
1.4582 |
|
0.1 |
0.005 |
2.606 |
2.6064 |
|
0.2 |
0.005 |
2.593 |
2.5926 |
|
0.3 |
0.005 |
2.568 |
2.5679 |
|
0.4 |
0.005 |
2.526 |
2.5262 |
|
0.5 |
0.005 |
2.456 |
2.456 |
|
0.6 |
0.005 |
2.342 |
2.3416 |
|
0.7 |
0.005 |
2.156 |
2.1564 |
|
0.8 |
0.005 |
1.876 |
1.8762 |
|
0.9 |
0.005 |
1.484 |
1.4839 |
|
0.1 |
0.001 |
2.723 |
2.723 |
|
0.2 |
0.001 |
2.709 |
2.7088 |
|
0.3 |
0.001 |
2.683 |
2.6827 |
|
0.4 |
0.001 |
2.638 |
2.6382 |
|
0.5 |
0.001 |
2.563 |
2.5628 |
|
0.6 |
0.001 |
2.44 |
2.4396 |
|
0.7 |
0.001 |
2.24 |
2.2397 |
|
0.8 |
0.001 |
1.938 |
1.9376 |
|
0.9 |
0.001 |
1.516 |
1.5161 |
|
0.1 |
0 (Sharp Edge) |
2.809 |
2.8093 |
|
0.2 |
0 (Sharp Edge) |
2.795 |
2.7949 |
|
0.3 |
0 (Sharp Edge) |
2.768 |
2.7679 |
|
0.4 |
0 (Sharp Edge) |
2.721 |
2.7213 |
|
0.5 |
0 (Sharp Edge) |
2.642 |
2.6422 |
|
0.6 |
0 (Sharp Edge) |
2.512 |
2.5124 |
|
0.7 |
0 (Sharp Edge) |
2.302 |
2.3016 |
|
0.8 |
0 (Sharp Edge) |
1.983 |
1.9832 |
|
0.9 |
0 (Sharp Edge) |
1.54 |
1.5398 |
Graphical Comparison of Formula:
Commentary: The published 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)