### Trapezoidal Notch

```Chapter Five
Fluid Measurement
OBJECT
At the end of this chapter the student should
be able to:
 Describe the importance of flow sensing and its
problem
 Describe the principle of operation of different
flow meter
 Describe the constructional and mean aspects
of differential meter.
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WHY IT IS IMPORTANT?
Almost all practical fluids engineering problems
are associated with the need for an accurate flow
measurement, since sometimes the material
transport from place to other entails payment of
financial dues
 There is a need to measure :
 local properties (velocity, pressure, temperature,
density, viscosity, turbulent intensity),
 integrated properties (mass flow and volume flow),
 global properties (visualization of the entire flow
field).
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WHAT WE MEASURE
Liquid
 Gases
 Multiphase fluid
 Solids
Custody Transfer Measuring System
Meter measurement system
Tank measuring system
Measuring system by balance
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LOCAL VELOCITY MEASURMENTS
1. Trajectory of floats or neutrally buoyant particles
 2. Rotating mechanical devices
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a. Cup anemometer
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b. Savonius rotor c. Propeller meter d. Turbine
meter
 3. Pitot-static tube
 4. Electromagnetic current meter
 5. Hot wires and hot films
 6. Laser-doppler anemometer (LDA)
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VOLUME-MASS MEASURMENTS
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These devices split into two classes:
mechanical instruments
1. Mass measurement
a. Weighing tanks
b. Tilting traps
2. Volume measurement
a. Volume tanks
b. Reciprocating pistons
c. Rotating slotted rings
d. Nutating disk
1. Bernoulli-type devices
a. Thin-plate orifice
b. Flow nozzle c. Venturi tube
2. Friction-loss devices
a. Capillary tube
b. Porous plug
There are other widely used meters operate on different physical principles:
1. Turbine meter
2. Vortex meter
3. Ultrasonic flowmeter
TYPES OF FLOW MEASURING
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The flow measurements can be classified into:
Obstructive Device
Differential pressure flow meter like:
Venture, orifice, pitot tube
Rotameter
Turbines
Non-obstructive
Electormegnatic
Ultrasonic
Cross-correlation
DIFFERENTIAL PRESSURE FLOW METER
Its widely used to measure the liquid and gas
 The principle is that a restriction is placed in
the pipe and the differential pressure
developed across the restriction is measured
 The differential pressure output is calibrated
in terms of volumetric flow rate (not mass this
will need density)
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1- ORIFICE METER
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The primary element of an orifice meter is
simply a flat plate containing a drilled (hole)
located in a pipe perpendicular to the
direction of fluid flow
Or
2- VENTURI METER
Or
These equations for venture and orifice are valid
for
 Turbulent flow
 Incompressible flow
 For gases : additional expansibility factor
 The values of discharge coeffiecient depends on
 Type of flow measurement; venture and orifice
 Diameter ratio
 Reynold number
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CHARACTERISTICS OF DIFFERENTIAL PRESSURE
MEASUREMENT
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No moving parts, cheap, maintainable
Well established, calibration available
Nonlinear relationship, so its not use for low pressure
since the pressure has square root with velocity
Discharge coefficient changes with wear, flow
distribution
Generally applicable for clean fluids
Installation constraints (for straight pipe not elbow)
COMPARISON OF VENTURE AND ORIFICE
Venturi
orifice
Expensive but offer good
accuracy
Least expensive
Long working life and almost no
maintenance
Low working life due to wear in
the edge
Can measure flow for fluid with
suspended solid
Used for clean fluid, can be used
for dilute slurries
High rang
3- PITOT TUBE
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The Pitot tube is used to measure the local velocity at a given point in the
flow stream and not the average velocity in the pipe or conduit.
One tube, the impact tube, has its opening normal to the direction of flow
and the static tube has its opening parallel to the direction of flow.
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The fluid flows into the opening at point 2, pressure builds up, and
then remains stationary at this point, called “Stagnation Point”. The
difference in the stagnation pressure (impact pressure) at this point
(2) and the static pressure measured by the static tube represents
the pressure rise associated with the direction of the fluid.
where, Cp: dimensionless coefficient to take into account deviations
from Bernoulli’s equation and general varies between about 0.98 to
1.0.
The first method, the velocity is measured at
the exact center of the tube to obtain umax.
then by using the Figure, the average velocity
can be obtained.
 The second method, readings are taken at
several known positions in the pipe cross
section and then a graphical or numerical
integration is performed to obtain the average
velocity, from the following equation;
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4 - THE NOZZLE
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The nozzle is similar to the orifice meter other than that
it has a converging tube in place of the orifice plate, as
shown in below. The velocity of the fluid is gradually
increased and the contours are so designed that almost
frictionless flow takes place in the converging portion;
the outlet corresponds to the vena contracta on the
orifice meter. When the ratio of the pressure at the
nozzle exit to the upstream pressure is less than the
critical pressure ratio ωc, the flow rate is independent of
the downstream pressure and can be calculated from
the upstream pressure alone.
Nozzle
Expensive
Long working life and almost
no maintenance
Generally Used to measure
steam
High discharge coefficient =
0.99
orifice
Least expensive
Low working life due to wear
in the edge
Used for clean fluid, can be
used for dilute slurries
Low discharge coefficient=
0.62
since it has no diverging cone
5 VARIABLE AREA METERS – ROTAMETERS
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In the Rotameter the drop in pressure is constant and the flow rate is function of
the area of constriction. When the fluid is flowing the float rises until its weight
is balanced by the up thrust of the fluid.
Force balance on the float
Gravity force = up thrust force +(drag force)Pressure forec
Vf ρf g = Vf ρg + (–ΔP) Af
A1 : cross-section area of the tube when the float arrived.
A2 : cross-section area of the annulus (flow area).
EXAMPLES
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1- A horizontal Venturi meter is used to measure the flow rate of
water through the piping system of 20 cm I.D, where the diameter of
throat in the meter is d2 = 10 cm. The pressure at inlet is 17.658
N/cm2 gauge and the vacuum pressure of 35 cm Hg at throat. Find
the discharge of water. Take Cd = 0.98.
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2- A Venturi meter is to be fitted to a 25 cm diameter pipe, in which
the maximum flow is 7200 lit/min and the pressure head is 6 m of
water. What is the maximum diameter of throat, so that there is nonnegative head on it?
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3- A (30cm x 15cm) Venturi meter is provided in a vertical pipe-line
carrying oil of sp.gr. = 0.9. The flow being upwards and the difference
in elevations of throat section and entrance section of the venture
meter is 30 cm. The differential U-tube mercury manometer shows a
gauge deflection of 25 cm. Take Cd = 0.98 and calculate: i-The discharge of oil
Ii-The pressure difference between the entrance and throat sections.
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EXAMPLES
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4- An orifice meter consisting of 10 cm diameter orifice in a 25 cm
diameter pipe has Cd = 0.65. The pipe delivers oil of sp.gr. = 0.8. The
pressure difference on the two sides of the orifice plate is measured
by mercury oil differential manometer. If the differential gauge is 80
cm Hg, find the rate of flow.
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5- Water flow through an orifice meter of 25 mm diameter situated in
a 75 mm diameter pipe at a rate of 300 cm3/s, what will be the
difference in pressure head across the meter μ = 1.0 mPa.s.
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6- Water flow at between 3000-4000 cm3/s through a 75 mm
diameter pipe and is metered by means of an orifice. Suggest a
suitable size of orifice if the pressure difference is to be measured
with a simple water manometer. What approximately is the pressure
difference recorded at the maximum flow rate? Cd = 0.6.
EXAMPLES
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7- A rotameter tube of 0.3 m long with an internal diameter of 25 mm at the
top and 20 mm at the bottom. The diameter of float is 20 mm, its sp.gr. is
4.8 and its volume is 6 cm3. If the coefficient of discharge is 0.7, what will
be the flow rate water when the float is half way up the tube?
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8- A Pitot tube is inserted in the pipe of 30 cm I.D. The static pressure head
is 10 cm Hg vacuum, and the stagnation pressure at center of the pipe is
0.981 N/cm2 gauge. Calculate the discharge of water through the pipe if
u/umax = 0.85. Take Cp = 0.98.
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9- A Pitot tube is placed at a center of a 30 cm I.D. pipe line has one orifice
pointing upstream and other perpendicular to it. The mean velocity in the
pipe is 0.84 of the center velocity (i.e. u/ux =0.94). Find the discharge
through the pipe if: i-The fluid flow through the pipe is water and the pressure difference
between orifice is 6 cm H2O.
Ii-The fluid flow through the pipe is oil of sp.gr. = 0.78 and the reading
manometer is 6 cm H2O. Take Cp = 0.98.
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6- THE NOTCH OR WEIR
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It is an obstruction in the channel that causes the liquid to back up behind it
and to flow over it or through it. By measuring the height of upstream water
surface, the rate of flow is determined. The velocity with which the liquid
leaves depends on its initial depth below the surface.
Many shapes of notch are available of which three shapes are given here as
follows,
6-1 Rectangular Notch
To prove this equation applies Bernoulli’s equation between points M and N as
shown in Figure;
The cross sectional area of flow at point M
is larger than that at notch (point N), then (uM ≈ 0)
PM = PN = Po atmospheric pressure
The discharge will be:
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6- THE NOTCH OR WEIR
6-2 Triangular Notch
 A triangular notch
is also called a V-notch.
 H: height of liquid above
base of the apex of the notch.
 θ: Angle of the notch.
 tan (θ/2) = x / H = x' / (H-h)
 The width of the notch at liquid surface = 2x = 2H tan(θ/2)
 The width of the strip = 2x' = 2(H-h) tan(θ/2)
 The area of the strip = 2x' dh = 2(H-h) tan(θ/2)dh
 The discharge is:
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6- THE NOTCH OR WEIR
6-3 Trapezoidal Notch
 A trapezoidal notch is
a combination of a
rectangular notch
and triangular notch as shown in Figure;
 Discharge over the trapezoidal notch,
Q=[Discharge over the rectangular notch
+ Discharge over the triangular notch]
 The discharge is
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EXAMPLES
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10- A rectangular notch has a discharge of 21.5 m3/min,
when the head of water is half the length of the notch. Find
the length of the notch where Cd = 0.6.
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11- A rectangular channel 1.5 m wide is used to carry 0.2
m3/s water. The rate of flow is measured by placing a 90º Vnotch weir. If the maximum depth of water is not to exceed
1.2 m, find the position of the apex of the notch from the
bed of channel. Cd = 0.6.
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12- A trapezoidal notch 120 cm wide at top and 45 cm at
the bottom has 30 cm height. Find the discharge through the
notch, if the head of water is 22.5 cm. Cd1 = Cd2 = 0.6.
NON-OBSTRACTIVE DEVICES
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Ultrasonic flowmeters.
Two examples are shown. The pulse-type flowmeter. Upstream piezoelectric
transducer A is excited with a short sonic pulse which propagates across the
flow to downstream transducer B. The arrival at B triggers another pulse to
be created at A, resulting in a regular pulse frequency f
The same process is duplicated in the reverse direction from B to A, creating
frequency . The is proportional to the flow rate.
The doppler type arrangement, where sound waves from transmitter T are
scattered by particles or contaminants in the flow to receiver R. Comparison
of the two signals reveals a doppler frequency shift which is proportional to
the flow rate. Ultrasonic meters are nonintrusive and can be directly
attached to pipe flows in the field . Their quoted uncertainty of 1 to 2
percent can rise to 5 percent or more due to irregularities in velocity profile,
fluid temperature, or Reynolds number.
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Hot-wire anemometer. A very fine wire (d 0.01 mm or less)
heated between two small probes, It is ideally suited to
measure rapidly fluctuating flows such as the turbulent
boundary layer. The idea dates back to work by L. V. King in
1914 on heat loss from long thin cylinders. If electric power
is supplied to heat the cylinder, the loss varies with flow
velocity across the cylinder according to King’s law
Because of its frailty, the hot wire is not suited to liquid flows,
whose high density and entrained sediment will knock the
wire right off. A more stable yet quite sensitive alternative for
liquid-flow measurement is the hot-film anemometer . A thin
metallic film, usually platinum, is plated onto a relatively
thick support which can be a wedge, a cone, or a cylinder.
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Laser-doppler anemometer. In the LDA a laser beam provides highly
focused, coherent monochromatic light which is passed through the flow.
When this light is scattered from a moving particle in the flow, a stationary
observer can detect a change, or doppler shift, in the frequency of the
scattered light. The shift f is proportional to the velocity of the particle.
There is essentially zero disturbance of the flow by the laser.
The advantages of the LDA are as follows:
No disturbance of the flow
High spatial resolution of the flow field
Velocity data that are independent of the fluid thermodynamic properties
An output voltage that is linear with velocity
No need for calibration
The disadvantages are that both the apparatus and the fluid must be
transparent to light and that the cost is high
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