This document discusses various methods for measuring fluid flow, including positive displacement meters and flow obstruction meters. Positive displacement meters have high accuracy but require clean fluids. Common positive displacement meters described are the nutating disk meter, rotary vane meter, and lobed impeller meter. Flow obstruction meters use a pressure drop measurement to determine flow rate. Common obstruction meters discussed are the Venturi meter, orifice plate, and flow nozzle. Empirical equations are provided for calculating flow rate using these various meter types. Examples are included to demonstrate flow rate calculations.

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

5. Nutating-disk meter
Youtube
5

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

7. Rotary-vane meter
Youtube
7

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

9. Lobed-impeller meter
Youtube
9

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

35. 6.4 Rotameters
37

36. 6.4 Rotameters
38

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

40. 6.5 Turbine Meters
42

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