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- 1. Engineering Measurements Chapter 6: Flow Measurement Naief Almalki, PhD
- 2. 6.1 Introduction • The measurement of fluid flow is important in applications ranging from measurements of blood-flow rates in a human artery to the measurement of the flow of liquid oxygen in a rocket. • It is usually required to plot the result and predict the relationship between these two variables. • The selection of the proper instrument for a particular application is governed by many variables, including cost. • It is easy to see how a small error in flow measurement on a large natural gas or oil pipeline could make a difference of thousands of dollars over a period of time. 2
- 3. 6.1 Introduction • Flow rate can be expressed in terms of a flow volume per unit time, known as the volume flow rate, or as a mass flow per unit time, known as the mass flow rate. • Flow rate is expressed in both volume and mass units of varying sizes. • Some commonly used terms are: 3
- 4. 6.2 Positive-displacement meters • In positive-displaced meter, the volume displaced on each cycle is known with great accuracy, allowing either the volume flow rate or the total volume passed to be determined. • Positive displacement meters have the advantages of high accuracy (from 0.2% to 2%), a very wide range of flow rate, the ability to retain accuracy at very low rates or during on/off flow conditions, and a general insensitivity to the viscosity or velocity profile of the fluid. • These meters have the disadvantage of requiring very clean fluids. 4
- 6. Nutating-disk meter • This meter operates on the nutating-disk principle. • Water enters the left side of the meter and strikes the disk, which is eccentrically mounted. • In order for the fluid to move through the meter the disk must “wobble” or nutate about the vertical axis since both the top and bottom of the disk remain in contact with the mounting chamber. • A partition separates the inlet and outlet chambers of the disk. As the disk nutates, it gives direct indication of the volume of liquid which has passed through the meter. • The indication of the volumetric flow is given through a gearing and register arrangement which is connected to the nutating disk. • The nutating-disk meter may give reliable flow measurements within 1%, over an extended period of time. 6
- 8. Rotary-vane meter • Another type of positive-displacement device is the rotary-vane meter. • The vanes are spring-loaded so that they continuously maintain contact with the casing of the meter. • A fixed quantity of fluid is trapped in each section as the eccentric drum rotates, and this fluid eventually finds its way out the exit. • An appropriate register is connected to the shaft of the eccentric drum to record the volume of the displaced fluid. • The uncertainties of rotary-vane meters are of the order of 0.5%, and the meters are relatively insensitive to viscosity since the vanes always maintain good contact with the inside of the casing. 8
- 10. Lobed-impeller meter • The lobed-impeller meter may be used for either gas- or liquid-flow measurements. • The impellers and case are carefully machined so that accurate fit is maintained. In this way the incoming fluid is always trapped between the two rotors and is conveyed to the outlet as a result of their rotation. • The number of revolutions of the rotors is an indication of the volumetric flow rate. 10
- 11. Example • A lobed-impeller flowmeter is used for measurement of the flow of nitrogen at 20 psia and 100◦F. The meter has been calibrated so that it indicates the volumetric flow with an accuracy of ± one-half of 1 percent from 1000 to 3000 cfm. The uncertainties in the gas pressure and temperature measurements are ±0.025 psi and ±1.0◦F, respectively. Calculate the uncertainty in a mass flow measurement at the given pressure and temperature conditions. 11
- 12. Solution: 𝑢 ሶ 𝑚 ሶ 𝑚 = 𝑢𝑄 𝑄 2 + 𝑢𝑃 𝑃 2 + 𝑢𝑇 𝑇 2 ൗ 1 2 𝑢 ሶ 𝑚 ሶ 𝑚 = 0.5 ∗ 3000 100 3000 2 + 0.025 20 2 + 1 100 2 ൗ 1 2 = 1.125% Note that the uncertainties in the pressure and temperature measurements do not appreciably influence the overall uncertainty in the mass flow measurements. 12
- 13. 6.3 Flow-Obstruction Methods • Flow-obstruction meters sometimes called head meters because a head-loss or pressure-drop measurement is taken as an indication of the flow rate. • Consider one-dimension, steady, incompressible and frictionless flow, the continuity and Bernoulli equations can be written as: Eq. 1 Eq. 2 13
- 14. 6.3 Flow-Obstruction Methods • The volume flow rate for incompressible fluid can be written as: • Where C is an empirical discharge coefficient that relate ideal to actual flow rates as: • The E in Eq.3 is the velocity of approach factor which can be calculated as: Eq. 3 14
- 15. 6.3 Flow-Obstruction Methods • The volume flow rate for compressible fluid can be written as: Or 𝑄 = 𝐾𝐴0𝑌 Τ 2∆𝑃 𝜌1 • Where Y is the compressible adiabatic expansion factor. • Empirical coefficients, C and Y are standardized for each type of flow obstructions. Eq. 4 15
- 16. 6.3 Flow-Obstruction Methods • Three typical obstruction meters are shown: (a) Venturi (b) Flow nozzle (c) Orifice • The venturi offers the advantages of high accuracy and small pressure drop, while the orifice is considerably lower in cost. • Both the flow nozzle and the orifice have a relatively high permanent pressure drop. • Flow-rate calculations for all three devices are made on the basis of Eq.4 with appropriate empirical constants. 16
- 17. 6.3 Flow-Obstruction Methods The permanent pressure loss associated with flow through common obstruction meters. 17
- 18. Orifice Meter 18
- 19. Orifice Meter • An orifice meter consists of a circular plate having a central hole (orifice). • The plate is inserted into a pipe so as to effect a flow area change. • The orifice hole is smaller than the pipe diameter and arranged to be concentric with the pipe’s i.d. • The common square-edged orifice plate is shown. • Installation is simplified by housing the orifice plate between two pipe flanges. • With this installation technique any particular orifice plate is interchangeable with others of different 𝛽 value. • The simplicity of the design allows for a range of 𝛽 values to be maintained on hand at modest expense. 19
- 20. Orifice Meter • Eq.4 is used with values of 𝐴𝑜 and 𝛽 based on the orifice (hole) diameter, 𝑑𝑜. • The plate thickness should be between 0.005-0.02 of the pipe diameter. • Standard pressure tap locations include (1) flange taps where pressure tap centers are located 25.4 mm (1 in.) upstream and 25.4 mm (1 in.) downstream of the nearest orifice face, (2) d upstream and d/2 downstream of orifice face, and (3) vena contracta taps. Nonstandard tap locations always require in situ meter calibration. • The relative instrument systematic uncertainty in the discharge coefficient is 0.6 % of C for 0.2 ≤ 𝛽 ≤ 0.6 and 𝛽% of C for all 𝛽 > 0.6. • The relative instrument systematic uncertainty for the expansion factor is about 4 𝑝1 − 𝑝2 /𝑝1 % of Y. • Realistic estimates of the overall systematic uncertainty in estimating Q using an orifice meter are between 1% (high 𝛽) and 3% (low 𝛽) at high Reynolds numbers when using standard tables. 20
- 21. Orifice Meter 21
- 22. Venturi Meter 22
- 23. Venturi Meter • A venturi meter consists of a smooth converging (21±2 degrees) conical contraction to a narrow throat followed by a shallow diverging conical section. • The engineering standard venturi meter design uses either a 15-degree or 7-degree divergent section. • The meter is installed between two flanges intended for this purpose. • Pressure taps are located just ahead of the upstream contraction and at the throat. • Eq.4 is used with values for both 𝐴 and 𝛽 based on the throat diameter, 𝑑𝑜. 23
- 24. Venturi Meter • The quality of a venturi meter ranges from cast to precision-machined units. • The discharge coefficient varies little for pipe diameters above 7.6 cm (3 in.). • In the operating range 2 × 105 ≤ 𝑅𝑒𝑑1 ≤ 2 × 106 and 0.4 ≤ 𝛽 ≤ 0.75, a value of 𝐶 = 0.984 with a systematic uncertainty of 0.7% (95%) for cast units and 𝐶 = 0.995 with a systematic uncertainty of 1% (95%) for machined units should be used. • Typical values for the flow coefficient and expansion factor are shown. • An instrument systematic uncertainty of 4 + 100𝛽2 𝑝1 − 𝑝2 /𝑝1 % 𝑜𝑓 𝑌. • A venturi meter presents a much higher initial cost over an orifice plate • A venturi meter demonstrates a much smaller permanent pressure loss for a given installation. • This translates into lower system operating costs for the pump or blower used to move the flow. 24
- 25. Venturi Meter 25
- 26. Flow Nozzles 26
- 27. Flow Nozzles • A flow nozzle consists of a gradual contraction from the pipe’s inside diameter down to a narrow throat. It needs less installation space than a venturi meter and has about 80% of the initial cost. • Common forms are the ISO 1932 nozzle and the ASME long radius nozzle. • The long radius nozzle contraction is that of the quadrant of an ellipse, with the major axis aligned with the flow axis, as shown. • The nozzle is typically installed inline, but can also be used at the inlet to and the outlet from a plenum or reservoir or at the outlet of a pipe. • Pressure taps are usually located(1) at one pipe diameter upstream of the nozzle inlet and at the nozzle throat using either wall or throat taps, or (2) d and d/2 wall taps located one pipe diameter upstream and one-half diameter downstream of the upstream nozzle face. • The flow rate is determined from Eq.4 with values for 𝐴𝑜and 𝛽 based on the throat diameter. • Typical values for the flow coefficient and expansion factor are shown. • The relative instrument systematic uncertainty at 95% confidence for the discharge coefficient is about 2% of C and for the expansion factor is about 2 𝑝1 − 𝑝2 /𝑝1 of Y. 27
- 28. Flow Nozzles 28
- 29. Example • A 10-cm-diameter, square-edged orifice plate is used to meter the steady flow of 16℃ water through an 20-cm pipe. Flange taps are used and the pressure drop measured is 50 cm Hg. Determine the pipe flow rate. The specific gravity of mercury is 13.5. 29
- 30. Given. 𝑑1 = 20 𝑐𝑚 = 0.2 𝑚 𝑑0 = 10 𝑐𝑚 = 0.1 𝑚 𝐻 = 50 𝑐𝑚 𝐻𝑔 = 0.5 𝑚 𝐻𝑔 𝜇 = 0.00108 ൗ 𝑁. 𝑠 𝑚2 𝜌 = 999 ൗ 𝑘𝑔 𝑚3 Incompressible flow of a liquid 𝑌 = 1 Find. Volume flow rate, Q Solution. 𝛽 = 𝑑0 𝑑1 = 10 20 = 0.5 The pressure drop based on the manometer deflection is ∆𝑃 = 𝜌𝑔𝐻 = 13.5 × 999 × 9.81 × 0.5 = 66151.28 𝑃𝑎 The flow Reynolds number is estimated as 𝑅𝑒 = 𝜌𝑉𝑑1 𝜇 = 4𝜌𝑄 𝜋𝑑1𝜇 = 4 × 999 × 𝑄 𝜋 × 0.2 × 0.00108 We see that without information concerning Q, we cannot determine the Reynolds number, and so C cannot be determined explicitly. Instead, a trial-and-error iteration is undertaken: Guess a value for K (or for C) and iterate. A good start is to guess a value at a high value of Re. This is the flat region of the Figure, so choose a value of K=CE =0.625. So 𝑄 = 𝐾𝐴0𝑌 Τ 2∆𝑃 𝜌1 = 0.625 × 𝜋 4 0.12 × 1 2 × 66151.28 999 = 0.056 ൗ 𝑚3 𝑠 Next we have to test the guessed value for K to determine if it was correct. For this value of Q 𝑅𝑒 = 4𝜌𝑄 𝜋𝑑1𝜇 = 4 × 999 × 0.056 𝜋 × 0.2 × 0.00108 = 3.2 × 105 From the Figure, at this Reynolds number, K=0.625. The solution is converged, so we conclude that Q = 0.056 ൗ 𝑚3 𝑠 31
- 31. 32
- 32. Example • Air flows at 20℃ through a 6-cm pipe. A square-edged orifice plate with 𝛽 = 0.4 is chosen to meter the flow rate. A pressure drop of 250 cm H2O is measured at the flange taps with an upstream pressure of 93.7 kPa abs. Find the flow rate. 33
- 33. Given. 𝑑1 = 6 𝑐𝑚 = 0.06 𝑚 𝛽 = 0.4 𝐻 = 250 cm H2O = 2. 5 𝑚 H2O 𝑃1 = 93.7 kPa abs=93700 Pa abs 𝑇1 = 20 + 273 = 293 𝐾 𝜌H2O = 999 ൗ 𝑘𝑔 𝑚3 𝜇𝑎𝑖𝑟 = 1.802 × 10−5 (𝑎𝑡 20℃) Find. Volume flow rate, Q Solution. 𝛽 = 𝑑0 𝑑1 = 𝑑0 0.06 = 0.4 → 𝑑0 = 0.024 𝑚 The air density is found from the ideal gas equation of state 𝜌1 = 𝑃1 𝑅𝑇1 = 93700 287 × 293 = 1.114 ൗ 𝑘𝑔 𝑚3 The pressure drop based on the manometer deflection is ∆𝑃 = 𝜌𝑔𝐻 = 999 × 9.81 × 2.5 = 24500.5 𝑃𝑎 The pressure ratio for this gas flow, Τ ∆𝑃 𝑃1 = Τ 24500.5 93700 = 0.26 For pressure ratios greater than 0.1 the compressibility of the air should be considered. From the Figure, 𝑌 = 0.92 for 𝑘 = 1.4 (air) and a pressure ratio of 0.26. The flow Reynolds number is estimated as 𝑅𝑒 = 4𝜌𝑄 𝜋𝐷𝜇 = 4 × 1.114 × 𝑄 𝜋 × 0.06 × 1.802 × 10−5 Similar to the previous example, a trial-and-error iteration is needed: A good start is to guess a value at a high value of Re. This is the flat region of the Figure for 𝛽 = 0.4 , so choose a value of K=CE =0.61. So 𝑄 = 𝐾𝐴0𝑌 Τ 2∆𝑃 𝜌1 = 0.61 × 𝜋 4 0.0242 × 0.92 2 × 24500.4 1.114 = 0.053 ൗ 𝑚3 𝑠 Next we have to test the guessed value for K to determine if it was correct. For this value of Q 𝑅𝑒 = 4𝜌𝑄 𝜋𝑑1𝜇 = 4 × 1.114 × 0.053 𝜋 × 0.06 × 1.802 × 10−5 = 7 × 104 From the Figure, at this Reynolds number, K=0.6. So we conclude that Q = 0.053 ൗ 𝑚3 𝑠 35
- 34. 36
- 37. 6.4 Rotameters • The average velocity sensed by the float depends on its height in the tube and is given by 39
- 38. 6.4 Rotameters • The rotameter is a widely used insertion meter for volume flow rate indication. • The meter consists of a float within a vertical tube, tapered to an increasing cross- sectional area at its outlet. • Flow entering through the bottom passes over the float, which is free to move. • The equilibrium height of the float indicates the flow rate. • The operating principle of a rotameter is based on the balance between the drag force, FD, and the weight, W, and buoyancy forces, FB, acting on the float in the moving fluid. • It is the drag force that varies with the average velocity over the float. 40
- 39. 6.4 Rotameters • This flow rate is found by • where Aa(y) is the annular area between the float and the tube, and K1 is a meter constant. • Both the average velocity and the annular area depend on the height of the float in the tube. So the float’s vertical position is a direct measure of flow rate, which can be read from a graduated scale, electronically sensed with an optical cell, or detected magnetically. • Floats with sharp edges are less sensitive to fluid viscosity changes with temperature. A typical meter is with an instrument systematic uncertainty of 2% (95%) of flow rate. 41
- 41. 6.5 Turbine Meters • Turbine meters make use of angular momentum principles to meter flow rate. • A rotor is encased within a bored housing through which the fluid to be metered is passed. Its housing contains flanges or threads for direct insertion into a pipeline. • In principle, the exchange of momentum within the flow turns the rotor at a rotational speed that is proportional to the flow rate. • Rotor rotation can be measured in a number of ways. For example, a reluctance pickup coil can sense the passage of magnetic rotor blades, producing a pulse train signal at a frequency that is directly related to rotational speed. • This can be directly output as a TTL pulse train, or the frequency can be converted to an analog voltage. 43
- 42. 6.5 Turbine Meters • Turbine meters offer a low-pressure drop and very good accuracy. • A typical instrument’s systematic uncertainty in flow rate is 0.25% (95%). • The measurements are exceptionally repeatable making the meters good candidates for local flow rate standards. • However, their use must be restricted to clean fluids because of possible fouling of their rotating parts. • The turbine meter rotational speed is sensitive to temperature changes, which affect fluid viscosity and density. 44
- 43. 6.6 Flow Meters Calibration 45
- 44. 6.6 Flow Meters Calibration • In liquids, variations of a ‘‘catch-and-weigh’’ technique are often employed in flow provers. • One variation of the technique consists of a calibration loop with a catch tank as shown. • Tank A is a large tank from which fluid is pumped back to a constant head reservoir, which supplies the loop with a steady flow. • Tank B is the catch-and-weigh tank into which liquid can be diverted for an accurately determined period of time. • The liquid volume is measured, either directly using a positive displacement meter, or indirectly through its weight, and the flow rate deduced through time. • The ability to determine the volume and the uncertainty in the initial and final time of the event are fundamental limitations to the accuracy of this technique. • The ultimate limits of uncertainty (at 95%) in the flow rate of liquids are on the order of 0.03%. 46