Lecture No

Lecture No. 9MEASUREMENT OF FLUID FLOW9.1 General types of flow meters

Obstruction meter. Fluid meters that belong to this type indicate a flow by a change in pressure. Full bore meter (Venturi meter, orifice meter, flow nozzle) Insertion meter (Pitot tube)

Variable area meter. An area meter is one in which the pressure drop is constant and the reading is dependent upon a variable flow area. Rotameter

Others: Magnetic field, vortex shedding, turbine meter etc.

9.2 Venturi meter Venturi meter is a full-bore obstruction meter with converging and diverging sections. It consists of a tube with a constricted throat which causes the increase of fluid velocity at the expense of pressure. The throat is followed by a gradually diverging section where the velocity is decreased with an increase in pressure and slight friction losses. A schematic of Venturi meter is shown in Figure 9.1.

Figure 9.1. Schematic of a Venturi meter

Velocity at the throat (u2):

(9.1)Mass flow rate () and velocity in the main line (u1):

(9.2)

For incompressible fluid flow:

(9.3)where:

Cv is the Venturi coefficient;

(= ;

-(Pis the pressure drop, P1 P2;

(is the density of the fluid;

(His the differential head, ();

Yis a correction factor.

Important notes:

If ( < 0.25, then .

Values of Cv:

Cv = 0.98

for a well-designed orifice

Cv ( 0.98

for D1 = 2 to 8 inches

Cv ( 0.99

for D1 > 8 inches

Cv = 0.984

for Herschel type with Re > 200,000 Values of Y:

For incompressible fluids (i.e., liquids):Y = 1

For compressible fluids (i.e., gases):

(9.4)where:

See also Equation 10-21 and Figure 10-16 of Perrys ChE Handbook (7th ed.)

The permanent pressure loss across Venturi meter is dependent on ( and the discharge cone angle (() in the diverging section:( = 5 7o

10 to 15% of -(P

( > 15o

10 to 30% of -(P Disadvantage: expensive compared to the other head meters.9.3 Orifice meterA sharp- or square-edged orifice is a clean-cut square-edged hole with straight walls perpendicular to the flat upstream face of a thin plate faced crosswise of a channel.The stream issuing from such an orifice attains its minimum cross section (vena contracta) at a distance downstream of the orifice which varies with (.

Figure 9.2. Schematic of a sharp-edged orificeVelocity at the orifice:

(9.5)where:

Cois the orifice discharge coefficient

Yis a correction factor.

Other parameters are as defined in Section 9.2.

Important notes: Values of Co = f(Re, ():

(9.6)

Equation 9.6 has been plotted to give Figure 9.3 below (or Figure 10-20 of Perrys ChE Handbook, 7th ed.)

Figure 9.3. Coefficient of discharge for square-edged circular (Figure 10-20 of Perrys ChE Handbook, 7th ed.)

Values of Y:For compressible fluid (i.e., gases):

(9.7)

See also Figure 10-16 of Perrys ChE Handbook (7th ed.)

9.4 Pitot tube

A pitot tube is an insertion meter that measures local velocity (i.e., velocity at a point). Applications include finding the velocity of a moving craft such as a boat or an airplane. A similar device, called Pitot static tube shown in Figure 9.4, is used to measure the velocity at different radial positions in a pipe.

Figure 9.4. Pitot tube with sidewall static tap (Fig. 10-5 Perrys ChE Handbook, 7th ed.)

Velocity at A (i.e., where the tip is located), uo:

(9.8)where:

Cis the correction coefficient

Important notes:

Values of C:C = 1 ( 0.01

for simple Pitot tubes

C = 0.98 to 1

for Pitot static tubes

For gases at velocities > 200 ft s-1:

(9.9)

Generally, for compressible fluids:

(9.10) For Equations 9.9 and 9.10, PA and PB are impact and static pressures, respectively. If the Pitot static tube is inserted at the middle of a tube or pipe, the velocity measured is maximum (uo = umax). To determine the average velocity in the tube or pipe, we use Figure 9.5.

Figure 9.5. Velocity ratio versus Reynolds number for smooth circular pipes (Figure 10-7 of Perrys ChE Handbook, 7th ed.)

9.5 Rotameter

The rotameter (Figure 9.6) consists of a solid float or plummet that is free to move inside a gradually tapered vertical glass tube. The fluid flows upward and the flow rate is indicated by the equilibrium position reached by the float which can be read from the adjacent scale usually etched on the glass tube.

Balance of forces at equilibrium (i.e., steady-state):

(9.11)

where:

FD drag force resulting from form and skin friction for flow around the float

FBbuoyant force acting so as to raise the float

FGgravity force acting downward on a float

Vfvolume of the float

(fdensity of the float

(density of the fluid

Figure 9.6. Schematic representation of a rotameter

Energy balance between 1 and 2 (see Figure 9.6)

(9.12)where:

(Fsummation of drag or friction lossesContinuity equation

(9.13)

Combining equations 9.12 and 9.13

(9.14)

(9.15)Drag force (FD)

(9.16)

where:

-(Pfpressure drop acting on the top of the float

fraction of the maximum pressure drop (-(P) that is not recovered

Combining equations 9.11, 9.15 and 9.16

(9.17)

where:

rotameter coefficient

(rotameter tapers gradually)

(9.18)

Mass flow rate

(9.19)Important notes: Proper design of rotameter float will make CR constant over wide range of Re. For a constant-density fluid in a single rotameter, the terms within the square root symbol of equation 9.18 are practically constant and independent of flow rate; hence,

(9.20)

The preceding equation shows that the velocity is equal to a constant multiplied by the minimum cross section for flow (S2).

Disadvantage: expensive in large installations.9.6 Notches and WeirsNotchAn opening in the side of the tank or reservoir which extends above the surface of the fluid. It is used to measure discharge flow rate.

WeirA notch on a lager scale, usually found in rivers. It maybe sharp-crested but may also have a substantial depth in the direction of flow. It can be used to measure flow rate or raise water levels.

The General Weir Equation

With reference to Figure 9.7,

Velocity through the strip:

(9.21)

Discharge through the strip:

(9.22)

Figure 9.7. Elemental strip of flow through a notch

Rectangular weir

Figure 9.8. Rectangular weirb = B

(9.23)

(9.24)

V-notch weir

Figure 9.9. V-notch or triangular weir geometry

(9.25)

(9.26)

(9.27)QUESTIONS / PROBLEMS:

1. Explain how a Pitot tube measures the speed of a boat.

2. Explain the principle of variable area meters.

3. A Pitot tube is inserted into the center of an air duct 0.8 m in diameter. A pressure gage attached to the Pitot tube reads -(P = 9 N/m2. The Pitot tube coefficient is 0.982. Calculate the mass flow rate of air, at a temperature of 35oC and a pressure of 101 kPa.

4. A Venturi meter with an entrance diameter of 0.3m and a throat diameter of 0.2m is used to measure the volume of gas flowing through a pipe. The discharge coefficient of the meter is 0.96. Assuming the specific weight of the gas to be constant at 19.62 N/m3, calculate the volume flowing when the pressure difference between the entrance and the throat is measured as 0.06 m on a water U-tube manometer.5. A Venturi meter is used for measuring flow of water along a pipe. The diameter of the Venturi throat is two-fifths the diameter of the pipe. The inlet and throat are connected by water-filled tubes to a mercury U-tube manometer. The velocity of flow along the pipe is found to be m/s, where H is the manometer reading in m of mercury. Determine the loss of head between inlet and throat of the Venturi when H is 0.49 m. The relative density of mercury is 13.6.6. Brine (specific gravity 1.18) is flowing though a 80-mm pipe at a maximum rate of 0.015 m3/s. In order to measure the flow rate, a sharp-edged orifice, connected to simple U-tube manometer is to be 390 mm Hg. What size orifice should be installed?

7. A closed tank has a 0.025-m diameter orifice in one of its vertical sides. The tank contains oil to a depth of 0.61 m above the centre of the orifice and the pressure in the air space above the oil is maintained at 13780 N/m2 above atmospheric. Determine the discharge from the orifice. The coefficient of discharge of the orifice is 0.61 and the relative density of oil is 0.9.8. A Venturi meter is fitted in a horizontal pipe of 0.15 m diameter to measure the flow of water which may be anything up to 240 m3/h. The pressure head at the inlet for this flow is 18 m above atmospheric and the pressure head at the throat must not be less than 7 m below atmospheric. Between the inlet and the throat there is an estimated frictional loss of 10% of the difference in pressure head between these points. Calculate the minimum allowable diameter for the throat.9. Deduce an expression for the discharge of water over a right-angled sharp edged V-notch, given that the coefficient of discharge is 0.61. A rectangular tank 16 m by 6m has the same notch in one of its short vertical sides. Determine the time taken for the head, measured from the bottom of the notch, to fall from 15 cm to 7.5 cm.10. Develop a formula for the discharge over a 90-degree V-notch weir in terms of head above the bottom of the V. A channel conveys 300 liters/sec of water. At the outlet end there is a 90-degree V-notch weir for which the coefficient of discharge is 0.58. At what distance above the bottom of the channel should the weir be placed in order to make the depth in the channel 1.30 m? With the weir in this position what is the depth of water in the channel when the flow is 200 liters/sec?

1

2

2

1

FG

FD

FB

Flow

B

A

85

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