Chapter 9 FLUID MEASUREMENT

1

• Chapter 9FLUID MEASUREMENT

2

• 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.

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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.

4

• 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.,

5
Figure 9.1 Static-pressure-measuring devices
(a) piezometer opening, (b) static tube.
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• 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.

7
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 high-speed
  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.

8

• 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

9

• The equation reduces
• (9.2.1)
• (9.2.2)
• Practically, it is very difficult to read the
  height ?h 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 ?h,
  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).

10
Figure 9.2 Simple pitot tube.
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• 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?.

12
Figure 9.3 Velocity measurement (a) pitot tube
and piezometer opening, (b) pitot-static tube.
13

• 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.

14

• 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.

15
Figure 9.4 Velocity-measuring devices (a) Price
current meter for liquids (W. and L.E. Gurley)
(b) air anemometer (Taylor Instrument Co).
16

• 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 V20 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.

17

• 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.

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• 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.

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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 right-angled 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.

20
Figure 9.5 Orifice in a reservoir.
21

• 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)

22

• 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,
• (9.3.6)
• from which (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.

23

• 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, Vatx0, 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 y0gt2/2 . After t is eliminated in the two
  relations,
•   
•               
• With V2t determined by Eq.(9.3.1), the ratio
  VaVtCv is known.

24

• 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,

25
Figure 9.6 Momentum method for determination of
Cv and Cc       
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• 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.

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• 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 x04.76m
  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

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• and the velocity is expressed by
• Then
• The actual discharge Qa is
• With Eq. (9.3.7)

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• Hence, from Eq. (9.3.6),
•                
• The head loss, from Eq. (9.3.8), is
•              
• The power loss is

30

• The Borda mouthpiece (Fig. 9.7), a short,
  thin-walled tube about one diameter long that
  projects into the reservoir (re-entrant), 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
•                   

31
Figure 9.7 The Borda mouthpiece.
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• 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
  Bernoullis eqn 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 CcA2/A0,
• (9.3.9)
• After eliminating V1t,
• and by solving for V2t the result is

33
Figure 9.8 Orifice in a pipe.
34

• 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 CdCvCc. 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)

35

• 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.

36
Figure 9.9 VDI orifice and discharge
coefficients.
37

• 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 dt
  is Qdt, 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 dt and integrating
• After substitution for Q,
• For the special case of a tank with constant
  cross section,

38
Figure 9.10 Notation for falling head.
39

• 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, Cd0.65,
  compute the time, in seconds, to lower the
  surface from 2.5 to 1 m above the orifice.
• Solution
• and

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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.

41

• 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 V1D12V2D22,
• (9.4.2)
• Equation (9.4.1) can be solved for V2t,
• and
• (9.4.3)

42
Figure 9.12 Venturi meter.
43

• Introducing the velocity coefficient V2aCvV2t
  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)

44

• 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 kinetic-energy correction factors a1, a2 in
  the Bernoulli equation.
• The loss is about 10 to 15 percent of the head
  change between sections 1 and 2.

45
Figure 9.13 Coefficient Cv for venturi meters.
46

• 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 k1.40.

47
Figure 9.14 Expansion factors.
48

• 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 ?pp1-p2. The value coefficient C in Fig.
  9.15 is for use in Eq. (9.4.10).

49
Figure 9.15 ISA (VDI) flow nozzle and discharge
coefficients.
50

• 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 15C.
• Solution
• From the data given, S013.6, S11.0, R'0.25 m,
  A2p/4000.00785 m2, ? 999.1 kg/m3, µ 0.00114
  Pas. Substituting Eq. (9.4.11) into Eq. (9.4.7)
  gives

51

• 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.

52

• 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.

53

• 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.

54

• 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.

55
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
  broad-crested 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.

56
Figure 9.16 Sharp-crested rectangular weir.
Figure 9.17 Weir nappe without contractions.
57

• 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.

58

• 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)

59
Figure 9.18 Weirs (a) horizontal with end
contractions (b) V-notch weir.
60

• 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 vv(2gy) and the
  theoretical discharge is
• By similar triangles, x may be related to y

61

• After substituting for v and x
• Expressing L/H in terms of the angle f 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)

62

• 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.

63
Figure 9.19 Broad-crested weir.
64

• 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

65

• 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.

66

• 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.

67

• 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 (1)
• and

68

• 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.

69
Figure 9.20 Log-log plot of Q vs. H for V-notch
weir.
70

• 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.

71

• 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 current-meter 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.

72
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.

73
Figure 9.21 Turbulent fluctuations in direction
of flow.
74
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 t, 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.

75

• 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 2pr2N/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.

76
Figure 9.22 Concentric-cylinder viscometer.
77

• Referring to Fig. 9.22b,
• Integrating over the circular area of the disk
  and letting ?2pN/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.

78

• 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.

79
Figure 9.23 Determination of viscosity by flow
through a capillary tube.
80

• 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
  ?, ?p 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
  ?p?gh, QV/t when the terms that are the same
  regardless of the fluid are separated,

81
Figure 9.24 Schematic view of Saybolt
viscometer.
82

• Since µ/? 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.

83

• 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.