Chapter 9
FLUID MEASUREMENT
• Fluid measurements include the determination of pressure,
velocity, discharge shock waves, density gradients, turbulence,
and viscosity. There are many ways these measurements may
be taken, e.g., direct, indirect, gravimetric, volumetric
electronic, electromagnetic, and optical.
• Direct measurements for discharge consist in the determination
of the volume or weight of fluid that passes a section in a given
time interval. Indirect methods of discharge measurement
require the determination of head, difference in pressure, or
velocity at several points in a cross section and, with these,
computing the discharge.
• The most precise methods are the gravimetric or volumetric
determinations, in which the weight or volume is measured by
weigh scales or by a calibrated tank for a time interval that is
measured by a stopwatch.
9.1 PRESSURE MEASUREMENT
• The measurement of pressure is required in many devices that
determine the velocity of a fluid stream or its rate of flow,
because of the relation between velocity and pressure given by
the energy equation. The static pressure of a fluid in motion is
its pressure when the velocity is undisturbed by
the measurement.
• Figure 9.1a indicates one method of measuring static pressure,
the piezometer opening. When the flow is parallel, as indicated,
the pressure variation is hydrostatic normal to the streamlines;
hence, by measuring the pressure at the wall, the pressure at
any other point in the cross section can be determined.
• The piezometer opening should be small, with length of opening
at least twice the diameter, and should be normal to the surface,
with no burrs at its edges, because small eddies would form and
distort the measurement.
• For rough surfaces, the static tube (Fig. 9.1b) may be used. It
consists of a tube that is directed upstream with the end closed.
It has radial holes in the cylindrical portion downstream from
the nose.
• The flow is presumed to be moving by the openings as if it were
undisturbed. There are disturbances, however, due to both the
nose and the right-angled leg that is normal to the flow.
• The static tube should be calibrated, as it may read too high or
too low. If it does not read true static pressure, the discrepancy
normally varies as the square of the velocity of flow by the tube;
i.e.,
Figure 9.1 Static-pressure-measuring devices:
(a) piezometer opening, (b) static tube.
• Such tubes are relatively insensitive to the Reynolds number
and to Mach numbers below unity. Their alignment with the
flow is not critical, so that an error of but a few percent is to
be expected for a yaw misalignment of 15℃.
• The piezometric opening may lead to a bourdon gage, a
manometer, a micromanometer, of an electronic transducer.
The transducers depend upon very small deformations of a
diaphragm due to pressure change to create an electronic
signal.
• The principle may be that of a strain gage and a Wheatstone
bridge circuit, or it may rely on motion in a differential
transformer, a capacitance chamber, or the piezoelectric
behavior of a crystal under stress.
8.2 VELOCITY MEASUREMENT
• Since determining velocity at a number of points in a cross
section permits evaluating the discharge, velocity measurement
is an important phase of measuring flow. Velocity can be found
by measuring the time an identifiable particle takes to move a
known distance. This is done whenever it is convenient or
necessary.
• This technique has been developed to study flow in regions
which are so small that the normal flow would be greatly
disturbed and perhaps disappear if an instrument were
introduced to measure the velocity.
• A transparent viewing region must be made available, and by
means of a strong light and a powerful microscope the very
minute impurities in the fluid can be photographed with a highspeed motion-picture camera. From such motion pictures the
velocity of the particles, and therefore the velocity of the fluid
in a small region, can be determined.
• The pitot tube operates on such a principle and is one of the
most accurate methods of measuring velocity. In Fig. 9.2 a glass
tube or hypodermic needle with a right-angled bend is used to
measure the velocity v in an open channel.
• The tube opening is directed upstream so that the fluid flows
into the opening until the pressure builds up in the tube
sufficiently to withstand the impact of velocity against it.
Directly in front of the opening the fluid is at rest.
• The streamline through 1 leads to the point 2, called the
stagnation point, where the fluid is at rest, and there divides and
passes around the tube. The pressure at 2 is known from the
liquid column within the tube. Bernoulli's equation, applied
between points 1 and 2, produces
• The equation reduces
(9.2.1)
(9.2.2)
• Practically, it is very difficult to read the height Δh from a free
surface.
• The pitot tube measures the stagnation pressure, which is also
referred to as the total pressure.
• The total pressure is composed of two parts, the static pressure
h0 and the dynamic pressure Δh, expressed in length of a
column of the flowing fluid (Fig. 9.2). The dynamic pressure is
related to velocity head by Eq. (9.2.1).
Figure 9.2 Simple pitot tube.
•
Figure 9.3a illustrates one arrangement. Bernoulli's equation applied
from 1 to 2 is
(9.2.3)
•
The equation for the manometer, in units of length of water, is
•
•
Simplifying gives
(9.2.4)
•
Substituting for
in Eq. (9.2.3) and solving for v results in
(9.2.5)
•
The pitot tube is also insensitive to flow alignment, and an error of only
a few percent occurs if the tube has a yaw misalignment of less than
15℃.
Figure 9.3 Velocity measurement: (a) pitot tube and
piezometer opening, (b) pitot-static tube.
• The static tube and pitot tube may be combined into one
instrument, called a pitot-static tube (Fig. 9.3b). Analyzing this
system in a manner similar to that in Fig. 9.3a shows that the
same relations hold; Eq. (9.2.5) expresses the velocity, but the
uncertainty in the measurement of static pressure requires a
corrective coefficient C to be applied:
(9.2.6)
• A particular form of pitot-static tube with a blunt nose, the
Prandtl tube, has been so designed that the disturbances due to
nose and leg cancel, leaving C = 1 in the equation. For other
pitot-static tubes the constant C must be determined by
calibration.
• The current meter (Fig. 9.4a) is used to measure the velocity of
liquid flow in open channels. The cups are so shaped that the
drag varies with orientation, causing a relatively slow rotation.
• The number of signals in a given time is a function of the
velocity. The meters are calibrated by towing them through
liquid at known speeds. For measuring high-velocity flow a
current meter with a propeller as rotating element is used, as it
offers less resistance to the flow.
• Air velocities are measured with cup-type or vane (propeller)
anemometers Fig (9.4b) which drive generators indicating air
velocity directly or drive counters indicating the number of
revolutions.
• By so designing the vanes that they have very low inertia,
employing precision bearings and optical tachometers which
effectively take no power to drive them, anemometers can be
made to read very low air velocities. They can be sensitive
enough to measure the convection air currents which the
human body causes by its heat emission to the atmosphere.
Figure 9.4 Velocity-measuring devices: (a) Price current meter
for liquids (W. and L.E. Gurley); (b) air anemometer
(Taylor Instrument Co).
Velocity Measurement in Compressible Flow
• The pitot-static tube may be used for velocity determinations in
compressible flow.
• Applying Eq. (7.3.7) to points 1 and 2 of Fig. 9.3b with V2=0 gives
•
• The substitution of cp is from Eq. (7.1.8). Equation (7.1.17) then
gives
• The static pressure p1 may be obtained from the side openings of
the pitot tube, and the stagnation pressure may be obtained from
the impact opening leading to a simple manometer; or p2 - p1 may
be found from the differential manometer.
Positive-Displacement Meters
• One volumetric meter, the positive-displacement meter, has
pistons or partitions which are displaced by the flowing fluid
and a counting mechanism that records the number of
displacements in any convenient unit, such as liters.
• A common meter is the disk meter, or wobble meter used on
many domestic water-distribution systems. The disk oscillates
in a passageway, so that a known volume of fluid moves
through the meter for each oscillation.
• A stem normal to the disk operates a gear train which, in turn,
operates a counter. In good condition, these meters are accurate
to within 1 percent. When they are worn, the error may be very
large for small flows, such as those caused by a leaky faucet.
• The flow of natural gas at low pressure is usually measured by
a volumetric meter with a traveling partition. The partition is
displaced by gas inflow to one end of the chamber in which it
operates, and then, by a change in valving, it is displaced to the
opposite end of the chamber.
• Oil flow or high-pressure gas flow ina pipeline is frequently
measured by a rotary meter in which cups or vanes move about
an annular opening and displace a fixed volume of fluid for
each revolution. Radial or axial pistons may be so arranged
that the volume of a continuous flow through them is
determined by rotations of a shaft.
• Positive-displacement meters normally have no timing
equipment that measures the rate of flow. The rate of steady
flow may be determined with a stopwatch to record the time for
displacement of a given volume of fluid.
9.3 ORIFICES
Orifice in a Reservoir
• An orifice may be used for measuring the rate of flow out of a
reservoir or through a pipe. An orifice in a reservoir or tank
may be in the wall or in the bottom. It is an opening, usually
round, through which the fluid flows, as in Fig. 9.5.
• The area of the orifice is the area of the opening. The portion of
the flow that approaches along the wall cannot make a rightangled turn at the opening and therefore maintains a radial
velocity component that reduces the jet area.
• The cross section where the con traction is greatest is called the
vena contracta. The streamlines are parallel throughout the jet
at this section, and the pressure is atmospheric.
Figure 9.5 Orifice in a reservoir.
• The head H on the orifice is measured from the center of the
orifice to the free surface. The head is assumed to be held constant.
• Bernoulli's equation applied from a point 1 on the free surface to
the center of the vena contracta, point 2, with local atmospheric
pressure as datum and point 2 as elevation datum, neglecting
losses, is written
• Inserting the values gives
(9.3.1)
• The ratio of the actual velocity to the theoretical velocity is called
the velocity coefficient; that is,
and hence
(9.3.2-3)
• The ratio of jet area at vena contracta to area orifice is
symbolized by another coefficient called the coefficient of
contraction:
(9.3.4)
• The area at the vena contracta is CcA0. The actual discharge is
thus
(9.3.5)
• It is customary to combine the two coefficients into a discharge
coefficient Cd,
• from which
(9.3.6)
(9.3.7)
• There is no way to compute the losses between points 1 and 2;
hence, must be determined experimentally. It varies from 0.95
to 0.99 for the square-edged or rounded orifice.
• Trajectory method.
• By measuring the position of a point on the trajectory of the
free jet downstream from the vena contracta (Fig.9.5) the actual
velocity Va can be determined if air resistance is neglected. The
x component of velocity does not change; therefore, Vat=x0, in
which t is the time for a fluid particle to travel from the vena
contracta to point 3.
• The time for a particle to drop a distance y0 under the action of
gravity when it has no initial velocity in that direction is
expressed by y0=gt2/2 . After t is eliminated in the two relations,
• With V2t determined by Eq.(9.3.1), the ratio VaVt=Cv is known.
• Direct measuring of Va. With a pitot tube placed at the vena
contracta, the actual velocity Va is determined.
•
• Direct measuring of jet diameter. With outside calipers, the
diameter of jet at the vena contracta may be approximated.
This is not a precise measurement and in general is less
satisfactory than the other methods.
•
• Use of momentum equation. When the reservoir is small
enough to be suspended on knife-equation, as in Fig. 9.6, it is
possible to determine the force F that creates the momentum in
the jet. With the momentum equation,
Figure 9.6 Momentum method for determination
of Cv and Cc
• Losses in orifice Flow
• The head loss in flow through an orifice is determined by
applying the energy equation with a loss term for the distance
between points 1 and 2 (Fig. 9.5),
• Substituting the values for this case gives
(9.3.8)
• in which Eq. (8.3.3) has been used to obtain the losses in terms
of H and Cv or V2a and Cv.
Example 9.1
• A 75-mm-diameter orifice under a head of 4.88 m discharges
907.6 kg water in 32.6 s. The trajectory was determined by
measuring x0=4.76m for a drop of 1.22 m. Determine Cv, Cc, Cd,
the head loss per unit gravity force, and the power loss.
Solution
• The theoretical velocity V2t is
• The actual velocity is determined from the trajectory. The time
to drop 1.22 m is
• and the velocity is expressed by
• Then
• The actual discharge Qa is
• With Eq. (9.3.7)
• Hence, from Eq. (9.3.6),
• The head loss, from Eq. (9.3.8), is
• The power loss is
• The Borda mouthpiece (Fig. 9.7), a short, thin-walled tube
about one diameter long that projects into the reservoir (reentrant), permits application of the momentum equation, which
yields, one relation between Cv and Cd.
• The velocity along the wall of the tank is almost zero at all
points; hence, the pressure distribution is hydrostatic.
• The final velocity is V2a; the initial velocity is zero; and Qa is the
actual discharge. Then
• and
• Substituting for Qa and V2a and simplifying lead to
Figure 9.7 The Borda mouthpiece.
• Orifice in a Pipe
• The square-edged orifice in a pipe (Fig. 9.8) causes a contraction of
the jet downstream from the orifice opening. For incompressible
flow Bernoulli’s eq’n applied from section 1 to the jet at its vena
contracta, section 2, is
• The continuity equation relates V1t and V2t with the contraction
coefficient Cc=A2/A0,
(9.3.9)
• After eliminating V1t,
• and by solving for V2t the result is
Figure 9.8 Orifice in a pipe.
• Multiplying by Cv to obtain the actual velocity at the vena
contracta gives
• And, finally multiplying by the area of the jet, CcA0, produces
the actual discharge Q.
(9.3.10)
• In which Cd=CvCc. In terms of the gage difference R’, Eq.(9.3.10)
becomes
(9.3.11)
• Because of the difficulty in determining the two coefficients
separately, a simplified formula is generally used,
(9.3.12)
• or its equivalent,
(9.3.13)
• Values of C are given in Fig. 9.9 for the VDI orifice.
• By a procedure explained in the next section, Eq.(9.3.12) can be
modified by an expansion factor Y (Fig.9.14) to yield actual
mass rate of compressible (isentropic) flow.
(9.3.14)
• The location of the pressure taps is usually so specified that an
orifice can be installed in a conduit and used with sufficient
accuracy without performing a calibration at the site.
Figure 9.9 VDI orifice and discharge coefficients.
• Unsteady Orifice Flow from Reservoirs
•
•
In the orifice situations considered, the liquid surface in the reser-voir
has been assumed to be held constant.
The volume discharged from the orifice in time δt is Qδt, which must
just equal the reduction in volume in the reservoir in the same time
increment (Fig.9.10.). Equating the two expressions gives
•
Solving for δt and integrating
•
After substitution for Q,
•
For the special case of a tank with constant cross section,
Figure 9.10 Notation for falling head.
Example 9.2
• A tank has a horizontal cross-sectional area of 2 m2 at the
elevation of the orifice, and the area varies linearly with
elevation so that it is 1 m2 at a horizontal cress section 3 m
above the orifice. For a 100-mm-diameter orifice, Cd=0.65,
compute the time, in seconds, to lower the surface from 2.5 to 1
m above the orifice.
Solution
and
9.4 VENTURI METER, NOZZLE, AND
OTHER RATE DEVICES
Venturi Meter
• The venturi meter is used to measure the rate of flow in a pipe.
It is generally a casting (Fig 9.12) consisting of
– (1) an upstream section which is the same size as the pipe,
has a bronze liner, and contains a piezometer ring for
measuring static pressure;
– (2) a converging conical section;
– (3) a cylindrical throat with a bronze liner containing a
piezometer ring;
– (4) a gradually diverging conical region leading to a
cylindrical section the size of the pipe.
• A differential manometer is attached to the two piezometer
rings. The amount of discharge in incompressible flow is shown
to be a function of the manometer reading.
• The pressures at the upstream section and throat are actual
pressures, and the velocities from Bernoulli's equation are
theoretical velocities. When losses are considered in the energy
equation, the velocities are actual velocities.
• From Fig. 9.12
(9.4.1)
• With the continuity equation V1D12=V2D22,
(9.4.2)
• Equation (9.4.1) can be solved for V2t,
• and
(9.4.3)
Figure 9.12 Venturi meter.
• Introducing the velocity coefficient V2a=CvV2t gives
(9.4.4)
• After multiplying by A2, the actual discharge Q is determined
to be
(9.4.5)
• In units of length of water
• Simplifying gives
(9.4.6)
• By substituting into Eq.(9.4.5),
(9.4.7)
• The discharge depends upon the gage difference R‘ regardless
of the orientation of the venturi meter; whether it is horizontal,
vertical, or inclined, exactly the same equation holds.
• Experimental results for venturi meters are given in Fig. 9.13.
The coefficient may be slightly greater than unity for venturi
meters that are unusually smooth inside. This does not mean
that there are no losses ; it results from neglecting the kineticenergy correction factors α1, α2 in the Bernoulli equation.
• The loss is about 10 to 15 percent of the head change between
sections 1 and 2.
Figure 9.13 Coefficient Cv for venturi meters.
Venturi Meter for Compressible Flow
• The theoretical flow of a compressible fluid through a venturi
meter is substantially isentropic and is obtained from
Egs.(7.3.2), (7.3.6), and (7.3.7). When multiplied by Cv, the
velocity coefficient, it yields for mass flow rate
(9.4.8)
• Equation (9.4.5), when reduced to horizontal flow and modified
by insertion of an expansion factor, can be applied to
compressible flow
(9.4.9)
• Values of Y are plotted in Fig. 9.14 for k=1.40.
Figure 9.14 Expansion factors.
Flow Nozzle
• The ISA (Instrument Society of America) flow nozzle
(originally the VDI flow nozzle) is shown in Fig. 9.15. It has no
contraction of the jet other than that of the nozzle opening;
therefore, the contraction coefficient is unity.
• Equations (9.4.5) and (9.4.7) hold equally well for the flow
nozzle. For a horizontal pipe (h = 0), Eq. (9.4.5) may be written
(9.4.10)
in which
(9.4.11)
• and Δp=p1-p2. The value coefficient C in Fig. 9.15 is for use in
Eq. (9.4.10).
Figure 9.15 ISA (VDI) flow nozzle and discharge coefficients.
Example 9.4
• Determine the flow through a 150-mm-diameter water line that
contains a 100 mm-diameter flow nozzle. The mercury-water
differential manometer has a gage difference of 2501 mm.
water temperature is 15°C.
Solution
• From the data given, S0=13.6, S1=1.0, R'=0.25 m,
A2=π/400=0.00785 m2, ρ = 999.1 kg/m3, μ = 0.00114 Pa·s.
Substituting Eq. (9.4.11) into Eq. (9.4.7) gives
• From Fig. 9.15, for A2/A1= (10/15)2 = 0.444, assume that the
horizontal of the curves applies. Hence, C = 1.056; then
compute the flow and the Reynolds number.
• The chart shows the value of C to be correct; therefore, the
discharge is 65.2 L/s.
Elbow Meter
• The elbow meter for incompressible flow is one or the simplest
flow-rate-measuring devices. Piezometer openings on the inside
and on the outside or the elbow are connected to a differential1
manometer.
• Because of centrifugal force at the bend, the difference in
pressures is related to the discharge. A straight calming length
should precede the elbow, and, for accurate results, the meter
should be calibrated in place.
• As most pipelines have an elbow, it may be used as the meter.
After calibration the results are as reliable as with a venturi
meter or a flow nozzle.
Rotameter
• The rotameter is a variable-area meter that consists or an
enlarging transparent tube and a metering "float " (actually
heavier than the liquid) that is displaced upward by the
upward how or fluid through the tube.
• The tube is graduated to read the flow directly. Notches in the
float cause it to rotate and thus maintain a central position in
the tube.
• The greater the flow, the higher the position the float assumes.
Electromagnetic Flow Devices
• If a magnetic field is set up across a nonconducting tube and a
conducting fluid flows through the tube, an induced voltage is
produced across the flow which can be measured if electrodes
are embedded In the tube walls.
• The voltage is a linear function or the volume rate passing
through the tube. Either an ac or a dc field may be used, with a
corresponding signal generated at the electrodes.
• A disadvantage of the method is the small signal received and
the large amount of amplification needed. The device has been
used to measure the flow in blood vessels.
9.5 WEIRS
• Open-channel flow may be measured by a weir, which is an
obstruction in the channel that causes the liquid to back up
behind it and flow over it or through it. By measuring the
height or upstream liquid surface, the rate or flow is
determined.
• Weirs constructed from a sheet or metal or other material so
that the jet, or nappe springs free as it leaves the upstream face
are called sharp-crested weirs. Other weirs, such as the broadcrested weir, support the flow in a longitudinal direction.
• The sharp-crested rectangular weir (Fig. 9.16) has a horizontal
crest. The nappe is contracted at top and bottom as shown. An
equation for discharge can be derived if the contractions are
neglected. Without contractions the flow appears as in Fig. 9.17.
The nappe has parallel streamlines with atmospheric pressure
throughout.
Figure 9.16 Sharp-crested rectangular weir.
Figure 9.17 Weir nappe without contractions.
• Bernoulli's equation applied between 1 and 2 is
• in which the velocity head at section 1 is neglected. By solving
for v gives
• The theoretical discharge Qt is
• in which L is the width of weir. Experiment shows that the
exponent of H is correct but the coefficient is too great.
• The contractions and losses reduce the actual discharge to
about 62 percent of the theoretical, or
(9.5.1)
• When the weir does not extend completely across the width of
the channel it has end contractions, illustrated in Fig. 9.18a.
• An empirical correction for the reduction of flow is
accomplished by subtracting 0.1H from L for each end
contraction. The weir in Fig. 9.16 is said to have its end
contractions suppressed.
• A correction may be added to the head,
(9.5.2)
Figure 9.18 Weirs: (a) horizontal with end contractions;
(b) V-notch weir.
• With this trial discharge, a value of V is computed, since
• For small discharges the V-notch weir is particularly
convenient. The contraction of the nappe neglected, and
theoretical discharge is computed (Fig. 9.18b) as follows.
• The velocity at depth y is v=√(2gy) and the theoretical
discharge is
• By similar triangles, x may be related to y
• After substituting for v and x
• Expressing L/H in terms of the angle φ of the V notch gives
• The exponent in the equation is approximately correct, but the
coefficient must be reduced by about 42 percent because of the
neglected contractions.
• An approximate equation for a 900 V-notch weir is
(9.5.3)
• The broad-crested weir (Fig. 9.19a) supports the nappe so that
the pressure variation is hydrostatic at section 2. Bernoulli's
equation applied between points 1 and 2 can be used to find the
velocity v2 at height z, neglecting the velocity of approach,
• In solving for v2,
• z drops out; hence, v2 is constant at section 2. For a weir of
width L normal to the plane or the figure, the theoretical
discharge is
(9.5.4)
• A Plot of Q as abscissa against the depth y as ordinate, for
constant H, is given in Fig. 9.19b.
Figure 9.19 Broad-crested weir.
• A gate or other obstruction placed at section 3 of Fig. 9.19a can
completely stop the flow by making y = H. Now, if small flow is
permitted to pass section 3 (holding H constant), the depth y
becomes a little less than H and the discharge is, for example,
as shown by point a on the depth-discharge curve.
• By further lifting of the gate or obstruction at section 3, the
discharge-depth relation follows the upper portion of the curve
until the maximum discharge is reached.
• By taking dQ/dy and with the result set equal to zero, for
constant H,
and solving for y gives
• Inserting the value of H, that is, 3y/2, into the equation for
velocity v2 gives
and substituting the value of y into Eq. (9.5.4) leads to
(9.5.5)
• Experiment show that, for a well-rounded upstream edge, the
discharge is
(9.5.6)
• which is within 2 percent of the theoretical value. The flow,
therefore, adjust itself to discharge at the maximum rate.
Example 9.5
• Tests on a 600 V-notch weir yield the following values of head H
on weir and discharge Q:
• By means of the theory of least squares, determine the
constants in Q = CHm for this weir.
Solution
• By taking the logarithm or each side of the equation
ln Q = ln C + m ln H
or
y = B + mx
• it is noted that the best values of B and m are needed for a
straight line through the data when plotted on log-log paper.
• By the theory of least squares, the best straight line through the
data points is the one yielding a minimum value of the sums of
the squares of vertical displacements of each point from the line;
or, from Fig. 9.20,
• where n is the number of experimental points. To minimize F,
∂F/∂B and ∂F/∂m are taken and set equal to zero, yielding two
equations in the two unknowns B and m, as follows:
• from which
and
(1)
or
(2)
• Solving Eqs. (1) and (2) for m gives
• These equations are readily solved by an electronic hand
calculator having the ∑ key, or a simple program may be
written for the digital computer.
• The answer for the data of this problem is m = 2.438, C =
0.7155.
Figure 9.20 Log-log plot of Q vs. H for V-notch weir.
Measurement of River Flow
• Daily records of the discharge of rivers over long periods or
time are essential to economic planning for utilization of their
water resources or protection against floods.
• The daily measurement of discharge by determining velocity
distribution over a cross section of the river is costly. To avoid
the cost and still obtain daily records, control sections are
established where the river channel is stable, i.e., with little
change in bottom or sides of the stream bed.
• The control section is frequently at a break in slope of the river
bottom where it becomes steeper downstream.
• A gage rod is mounted at the control section, and the elevation
of water surface is determined by reading the waterline on the
rod; in some installations float-controlled recording gages keep
a continuous record of river elevation.
• A gage height-discharge curve is established by taking currentmeter measurements from time to time as the river discharge
changes and plotting the resulting discharge against the gage
height.
• With a stable control section the gage height-discharge curve
changes very little, and current-meter measurements are
infrequent.
• Daily readings of gage height produce a daily record of the
river discharge.
9.6 MEASUREMENT OF TURBULENCE
• Turbulence is a characteristic of the flow. It affects the
calibration of measuring instruments and has an important
effect upon heat transfer, evaporation, diffusion, and many
other phenomena connected with fluid movement.
• Turbulence is generally specified by two quantities, the size and
the intensity of the fluctuations. In steady flow the temporal
mean velocity components at a point are constant. It these
mean values are u, v, w and the velocity components at an
instant are u, v, w, the fluctuations are given by u’, v’, w’, in
• The root mean square of measured values of the fluctuation
(Fig. 9.21) is a measure of the intensity of the turbulence.
Figure 9.21 Turbulent fluctuations in direction of flow.
9.7 MEASUREMENT OF VISCOSITY
• The treatment of fluid measurement is concluded with a
discussion of methods for determining viscosity.
• Viscosity may be measured in a number of ways: (1) by use of
Newton's law of viscosity, (2) by use or the Hagen-Poiseuille
equation, (3) by methods that require calibration with fluids of
known viscosity.
• By measurement of the velocity gradient du/dy and the shear
stress τ, in Newton's law of viscosity [Eq. (1.1.1)],
(9.7.1)
• the dynamic or absolute viscosity can be computed. This is the
most basic method, because it determines all other quantities in
the defining equation for viscosity.
• A schematic view of a concentric-cylinder viscometer is shown
in Fig. 9.22a. When the speed of rotation is N rpm and the
radius is r2, the fluid velocity at the surface of the outer
cylinder is 2πr2N/60. With clearance b
• If the torque due to fluid below the bottom of the inner cylinder
is neglected, the shear stress is
• Substituting into Eq. (9.7.1) and solving for the viscosity yields
(9.7.2)
• When the clearance a is so small that the torque contribution
from the bottom is appreciable, it can be calculated in terms of
the viscosity.
Figure 9.22 Concentric-cylinder viscometer.
• Referring to Fig. 9.22b,
• Integrating over the circular area of the disk and letting
ω=2πN/60 leads to
(9.7.3)
• The torque due to disk and cylinder must equal the torque T in
the torsion wire, so that
(9.7.4)
• in which all quantities except are known. The flow between
the surfaces must be laminar for Eqs. (9.7.2) to (9.7.4) to be
valid.
• Often the geometry of the inner cylinder is altered to eliminate
the torque which acts on the lower surface. If the bottom
surface of the inner cylinder is made concave, a pocket of air
will be trapped between the bottom surface of the inner
cylinder and the fluid in the rotating outer cup.
• A well-designed cup and a careful filling procedure will ensure
the condition whereby the torque measured will consist of that
produced in the annulus between the two cylinders and a
minute amount resulting from the action of the air on the
bottom surface.
• The measurement of all quantities in the Hagen-Poiseuille
equation, except μ, by a suitable experimental arrangement is
another basic method for determination of viscosity. A setup as
in Fig. 9.23 may be used.
Figure 9.23 Determination of viscosity by flow through
a capillary tube.
• The volume V of flow can be measured over a time t where the
reservoir surface is held at a constant level. This yields Q; and
by determining γ, Δp can be computed. Then with L and D
known, from Eq.(5.4.10a),
• An adaptation of the capillary tube for industrial purposes is
the Saybolt viscometer (Fig. 9.24). A short capillary tube is
utilized, and the time is measured for 60 cm3 of fluid to flow
through the tube under a falling head.
• This device measures kinematic viscosity, evident from a
rearrangement of Eq. (5.4.10a). When Δp=ρgh, Q=V/t; when the
terms that are the same regardless of the fluid are separated,
Figure 9.24 Schematic view of Saybolt viscometer.
• Since μ/ρ = v, the kinematic viscosity is
• A correction in the above equation is needed, which is of the
form C/t; hence,
• The approximate relation between viscosity and Saybolt
seconds is expressed by
• in which v is in stokes and t in seconds.
• For measuring viscosity there are many other industrial
methods that generally have to be calibrated for each special
case to convert to the absolute units.
• One consists of several tubes containing "standard " liquids of
known graduated viscosities with a steel ball in each of the
tubes.
• The flow of a fluid in a capillary tube is the basis for
viscometers of the Oswald-Cannon-Fenske, or Ubbelohde, type.
In essence, the viscometer is a U tube one leg of which is a fine
capillary tube connected to a reservoir above. The tube is held
vertically, and a known quantity of fluid is placed in the
reservoir and allowed to flow by gravity through the capillary.
• The limit is recorded for the free surface in the reservoir to fall
between two scribed marks.