Preface	6
	Contents	8
	Notations	15
	1 Introduction and History	19
	Abstract	19
	1.1 Introduction	19
	1.2 Theory	20
	1.2.1 Bernoulli's Theorem	20
	1.2.2 Method of Operation	21
	1.2.2.1 General	21
	1.2.2.2 Incompressible Flow	21
	1.2.2.3 Compressible Flow	22
	1.2.2.4 Equation for Practical Use	24
	1.3 Essential Requirements	25
	1.3.1 General	25
	1.3.2 With a Calibration in a Flowing Fluid	25
	1.3.3 Without a Calibration in a Flowing Fluid	26
	1.4 Introduction to Reynolds Number and Velocity Profile	26
	1.5 Pipe Roughness	29
	1.6 Accuracy	30
	1.7 Pressure Loss	31
	1.8 Standards	32
	1.9 Advantages and Disadvantages	32
	1.10 History	33
	1.11 Conclusions	40
	Appendix 1.A: Sextus Julius Frontinus	41
	References	46
	2 Orifice Design	50
	Abstract	50
	2.1 Introduction	50
	2.2 Orifice Plate	51
	2.2.1 General	51
	2.2.2 Flatness	53
	2.2.3 Surface Condition of the Upstream Face of the Plate	54
	2.2.4 Edge Sharpness	56
	2.2.5 Plate Thickness E and Orifice (Bore) Thickness e	58
	2.2.5.1 General	58
	2.2.5.2 Plate Thickness E	58
	2.2.5.3 Orifice (Bore) Thickness e	59
	2.2.5.4 Requirements	61
	2.2.6 Circularity	62
	2.3 The Pipe	62
	2.3.1 General	62
	2.3.2 Pressure Tappings	63
	2.3.2.1 General	63
	2.3.2.2 Flange and D and D/2 Tappings	63
	General	63
	Tapping Diameter	65
	Tapping Location	65
	2.3.2.3 Corner Tappings	67
	2.3.2.4 Number of Tappings	67
	2.3.3 Pipe Roughness	68
	2.3.3.1 Uniform Roughness	68
	2.3.3.2 Rough Pipes with a Smooth Portion Immediately Upstream of the Orifice	71
	2.3.3.3 Non-uniform Roughness	72
	2.3.4 Steps and Misalignment	74
	2.3.5 Eccentricity	76
	2.4 Dimensional Measurements	77
	2.5 Orifice Fittings	78
	2.6 Pressure Loss	79
	2.7 Reversed Orifice Plates	82
	2.8 Conclusions	84
	Appendix 2.A: Orifice Plates of Small Orifice Diameter	85
	2.A.1 Introduction and Test Work	85
	2.A.2 Conclusions	89
	References	90
	3 Venturi Tube Design	94
	Abstract	94
	3.1 Introduction	94
	3.2 Type	96
	3.2.1 General	96
	3.2.2 Machined Convergent (5.2.9, 5.5.3 and 5.7.2 of ISO 5167-4:2003)	97
	3.2.3 Rough-Welded Sheet-Iron Convergent (5.2.10, 5.5.4 and 5.7.3 of ISO 5167-4:2003)	97
	3.2.4 `As Cast' Convergent (5.2.8, 5.5.2 and 5.7.1 of ISO 5167-4:2003)	97
	3.2.5 Wider Range of Reynolds Number	98
	3.3 Angles, Pressure Loss and Truncation	98
	3.4 Dimensional Measurements	100
	3.5 Steps and Straightness	101
	3.6 Pressure Tappings	102
	3.7 Effects of Roughness and Reynolds Number	104
	3.8 High or Low Reynolds Number	104
	3.9 Conclusions	107
	Appendix 3.A:‚Effect of Roughness: Computational Fluid Dynamics	107
	3.A.1 General	107
	3.A.2 Venturi Tube Roughness	107
	3.A.2.1 Effect of Venturi Tube Roughness Height	107
	3.A.2.2 Effect of Reynolds Number	108
	3.A.2.3 Effect of Venturi Tube Roughness Type	108
	3.A.3 Pipe Roughness	109
	3.A.4 Effect of Rounding the Corner Between the Convergent Section and the Throat	110
	References	112
	4 General Design	114
	Abstract	114
	4.1 Introduction	114
	4.2 Impulse Lines	114
	4.2.1 General	114
	4.2.2 Tapping Locations and Slopes of Impulse Lines	116
	4.2.3 Density of the Fluids in Two Impulse Lines to Measure the Differential Pressure	118
	4.2.4 Length of Impulse Lines	121
	4.2.5 Blockage	122
	4.2.6 Damping of the Pressure Signal or Resonance	123
	4.3 Differential Pressure	123
	4.3.1 Differential-Pressure Transmitters	123
	4.3.2 Piezometer Rings	126
	4.4 Static Pressure	127
	4.5 Drain and Vent Holes (Through the Pipe Wall)	128
	4.6 Temperature	128
	4.6.1 General	128
	4.6.2 Temperature Correction from Downstream of the Flowmeter to Upstream of It	129
	4.6.3 Using a Densitometer	132
	4.6.4 Correction of Dimensions for Temperature	133
	4.7 Iteration	134
	4.8 Uncertainty	134
	4.9 Cavitation	135
	4.10 Diagnostics	135
	4.11 Mixtures	136
	4.12 Conclusions	137
	Appendix 4.A: Impulse Lines in Pulsating Flows	137
	Appendix 4.B:‚Measuring Low Differential Pressure at High Static Pressure	140
	4.B.1‚Introduction	140
	4.B.2‚The Problem	140
	4.B.3‚A Possible Solution	140
	References	141
	5 Orifice Discharge Coefficient	143
	Abstract	143
	5.1 Introduction	143
	5.2 History	144
	5.3 The EEC/API Database	147
	5.4 The Equation	150
	5.4.1 Introduction	150
	5.4.2 The Tapping Terms	150
	5.4.2.1 Introduction	150
	5.4.2.2 High Reynolds Number Tapping Terms	152
	Total Tapping Terms	152
	Upstream Term	152
	Downstream Term	154
	5.4.2.3 Low Reynolds Number Tapping Terms	156
	5.4.3 The C221E and Slope Terms	160
	5.4.4 A Term for Small Orifice Meters	162
	5.4.5 The Complete Equation	163
	5.5 Quality of Fit	164
	5.6 Equations and Comparison Between Them on the Basis of Deviations	171
	5.6.1 The Reader-Harris/Gallagher (RG) Equation as in API 14.3.1:1990	171
	5.6.2 The Stolz Equation in ISO 5167:1980	172
	5.6.3 Comparisons	172
	5.7 Uncertainty	175
	5.8 Conclusions	179
	Appendix 5.A: Better Options for Tapping Terms	179
	Appendix 5.B: Small Orifice Diameters Within the EEC/API Database	183
	Appendix 5.C: The PR14 Equation and an Equation in Terms of Friction Factor	186
	5.C.1 The PR14 Equation	186
	5.C.2 An Equation in Terms of Friction Factor	187
	Appendix 5.D: The Effect on the Discharge-Coefficient Equation of Changing the Expansibility-Factor Equation	188
	Appendix 5.E: Orifice Plates in Pipes of Small Diameter or with No Upstream or with No Downstream Pipeline or with No Upstream and No Downstream Pipeline	191
	5.E.1 Introduction	191
	5.E.2 Orifice Plates in Pipes of Small Diameter	191
	5.E.3 Orifice Plates with No Upstream or Downstream Pipeline	192
	5.E.4 Orifice Plates with No Upstream Pipeline	194
	5.E.5 Orifice Plates with No Downstream Pipeline	195
	Appendix 5.F: Lower Reynolds Number Limit for the Reader-Harris/Gallagher (1998) Equation	197
	References	199
	6 Orifice Expansibility Factor	203
	Abstract	203
	6.1 Introduction	203
	6.2 History and Theory	204
	6.3 The Database	205
	6.4 Analysis	206
	6.5 Theoretical Model	211
	6.6 Subsequent Work	214
	6.7 Conclusions	215
	Appendix 6.A: Data Taken with a Flow Conditioner 7D or 10D from the Orifice Plate	215
	References	216
	7 Venturi Tube Discharge Coefficient in High-Pressure Gas	218
	Abstract	218
	7.1 Introduction	218
	7.2 Experimental Work: Standard Shape	219
	7.2.1 Description of the Venturi Tubes	219
	7.2.2 Calibration in Water	220
	7.2.3 Calibration in Gas	221
	7.3 Interpretation and Analysis of Data	224
	7.3.1 Static-Hole Error	224
	7.3.2 Measurements of Static-Hole Error at High Tapping-Hole Reynolds Number	225
	7.3.3 Measurements of Static-Hole Error at Low Tapping-Hole Reynolds Number	226
	7.3.4 The Effect of Tapping Depth on Static-Hole Error	227
	7.3.5 The Effect of Tapping Shape on Static-Hole Error	229
	7.3.6 The Effect of a Burr or a Protruding Tapping on Static-Hole Error	229
	7.3.7 Analysis of the Gas Data in Sect. 7.2.3	229
	7.3.8 Conclusions to Sect. 7.3	230
	7.4 Improved Shape	232
	7.4.1 General	232
	7.4.2 Venturi Tube with Convergent Angle 10.5?	232
	7.4.3 The Discharge-Coefficient Equation for Venturi Tubes with Convergent Angle 10.5?	234
	7.5 Conclusions	236
	Appendix 7.A: Shape of Venturi Tubes: Tests at NEL	236
	7.A.1 Design	236
	7.A.2 Calibration in Water	239
	7.A.3 Calibration in Gas	239
	7.A.4 Analysis	239
	7.A.5 Conclusions on Shape from the 42033 Venturi Tubes	247
	7.A.6 Manufacture of Additional Venturi Tubes with 10.5? Convergent Angle and Sharp Corners	247
	7.A.7 Calibration of Additional Venturi Tubes in Water and in Gas	248
	Appendix 7.B: Depth of Tappings: Tests at NEL	252
	Appendix 7.C: Refitting the Data With Convergent Angle 10.5?	254
	References	257
	8 Installation Effects	259
	Abstract	259
	8.1 Introduction	259
	8.2 Upstream Straight Lengths	260
	8.2.1 General	260
	8.2.2 Definitions	262
	8.2.3 Orifice Plates	262
	8.2.3.1 History	262
	8.2.3.2 The Pattern of the Data	262
	8.2.3.3 The Straight Lengths in ISO 5167-2:2003	267
	8.2.4 Venturi Tubes	269
	8.2.4.1 Standard Venturi Tubes	269
	General	269
	Calibrations Downstream of a Contraction and an Expansion	270
	Calibrations Downstream of Bends	270
	Analysis	273
	8.2.4.2 Venturi Tubes with Convergent Angle 10.5?	274
	8.2.5 What to Do if a Case is not Covered in Table 3 of ISO 5167-2:2003/Table 1 of ISO 5167-4:2003	276
	8.2.5.1 General	276
	8.2.5.2 The Upstream Installation is a Combination of Fittings	276
	8.2.5.3 A Flow Conditioner Is Used	277
	General	277
	With an Orifice Plate	278
	With a Venturi Tube	282
	Damage to Flow Conditioners	284
	8.2.5.4 A Specific Test Is Done	284
	8.2.5.5 CFD is Carried Out	285
	8.2.5.6 Engineering Judgement is Employed	285
	8.3 Downstream Straight Length	286
	8.3.1 Orifice Plates	286
	8.3.2 Venturi Tubes	286
	8.4 Pulsations	287
	8.4.1 General	287
	8.4.2 Orifice Plates	288
	8.4.3 Venturi Tubes	288
	8.5 Conclusions	288
	Appendix 8.A: Swirl Decay	289
	References	289
	9 Nozzle Discharge Coefficient	295
	Abstract	295
	9.1 Introduction	295
	9.2 Manufacture	297
	9.2.1 General	297
	9.2.2 Pipework and Nozzles	298
	9.2.3 Nozzle Tappings	298
	9.2.4 Wall Tappings	300
	9.3 Data	300
	9.4 Wall-Tapping Data: Analysis	301
	9.5 Throat-Tapping Data: Initial Analysis	305
	9.6 Hot-Water (NMIJ Throat-Tapping) Data	309
	9.7 Throat-Tapping Data: Further Analysis	310
	9.7.1 General	310
	9.7.2 Analysis of NMIJ Data	312
	9.7.3 Application to NEL Data	313
	9.7.4 Analysis of NEL Data	315
	9.8 Conclusions	316
	References	317
	10 Orifice Plates with Drain Holes	319
	Abstract	319
	10.1 Introduction	319
	10.2 Experimental Work: Initial Data	322
	10.3 Experimental Work: Additional Data	327
	10.4 Analysis	330
	10.4.1 Bernoulli's Theorem	330
	10.4.2 Pressure Tapping Location for Flow Measurement Without Error	332
	10.4.3 An Equation for the Corrected Diameter	334
	10.4.4 Practical Equations for the Corrected Diameter	338
	10.5 Conclusions	339
	References	339
	11 Wet Gas	341
	Abstract	341
	11.1 Introduction	341
	11.2 Fundamental Equations	343
	11.2.1 General	343
	11.2.2 Laboratory Test Work	344
	11.2.3 Models for Field Use	344
	11.2.3.1 General	344
	11.2.3.2 Venturi Tube	345
	General	345
	de Leeuw Equation	345
	ISO/TR 11583:2012 Correlation	345
	11.2.3.3 Orifice Plate	346
	11.2.4 Methods to Obtain the Lockhart-Martinelli Parameter, X (Eq. 11.2)	347
	11.2.4.1 General	347
	11.2.4.2 Pressure Loss Ratio	348
	Venturi Tube	348
	Orifice Plate	349
	11.3 Venturi Tubes	350
	11.3.1 Over-Reading Equations	350
	11.3.1.1 Derivation of the ISO/TR 11583:2012 Correlation	350
	11.3.1.2 Comparison with the de Leeuw Equation	357
	11.3.1.3 Possible Improvement of the ISO/TR 11583:2012 Correlation	357
	11.3.2 Using Pressure-Loss Measurements	361
	11.3.3 Mixtures of Liquids	365
	11.4 Orifice Plates	367
	11.4.1 General	367
	11.4.2 Derivation of the Equations in ISO/TR 11583:2012	367
	11.4.3 Subsequent Work	371
	11.5 Conclusions	371
	Appendix 11.A: A Brief History of ISO/TR 11583	372
	Appendix 11.B: Dependence of the Wet-Gas Correlations for Venturi Tubes on Liquid Viscosity	375
	11.B.1 General	375
	11.B.2 Deviations from the ISO/TR 11583:2012 Correlation	375
	11.B.3 Deviations from the de Leeuw Equation	378
	11.B.4 Errors Using ISO/TR 11583:2012 with X Determined from the Pressure Loss Ratio	380
	11.B.5 Analysis	380
	11.B.6 Horizontal Tappings	388
	References	388
	12 Standards	390
	Abstract	390
	12.1 Introduction	390
	12.2 ISO Standards	391
	12.3 ISO/TC 30 Measurement of Fluid Flow in Closed Conduits	392
	12.3.1 General	392
	12.3.2 ISO/TC 30/SC 2 Pressure Differential Methods	392
	12.3.2.1 General	392
	12.3.2.2 Differential-Pressure Flow Measurement Standards: ISO 5167 Etc.	393
	12.3.2.3 ISO/TR 9464 Guidelines for Using ISO 5167	395
	12.3.2.4 ISO/TR 12767 Differential-Pressure Meters Departing from ISO 5167	395
	12.3.2.5 ISO/TR 15377 Differential-Pressure Meters Beyond the Scope of ISO 5167	396
	12.3.2.6 ISO/TR 3313 Pulsating Flow	396
	12.3.2.7 ISO/TR 11583 Wet Gas	397
	12.3.2.8 ISO 2186 Impulse Lines	397
	12.3.2.9 Priorities for the Future as Seen in 2014	397
	12.3.3 The TC Itself	398
	12.3.3.1 General	398
	12.3.3.2 Priorities for the Future as Seen in 2014	399
	12.4 AGA/API Standards	399
	12.5 Conclusions	400
	Appendix 12.A: The Standards of ISO/TC 30/SC 2	400
	References	401
	Index	402