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CHAPTER 4
FLUID-FLOW APPLICATIONS
(cont'd)

MEASUREMENT OF VELOCITY IN A FLUID


As shown in Fig. 4.1(c), a bent tube is inserted into a flowing stream of fluid and orientated so that the mouth of the tube faces directly into the flow. The pressure in the tube will give a measure of velocity head due to the flow. Such a tube is called a Pitot tube.
The pressure exerted by the flowing fluid on the mouth of the tube is balanced by the manometric head of fluid in the tube. In equilibrium, when there is no movement of fluid in the tube, Bernouilli's equation can be applied. For the Pitot tube and manometer we can write:

Z1g + v12/2 + P1/r1 = Z2g + v22/2 + P2/r1

in which subscript 1 refers to conditions at the entrance to the tube and subscript 2 refers to conditions at the top of the column of fluid which rises in the tube.

       Now,

                                  Z2 = Z + Z'


   
        taking the datum level at the mouth of the tube and letting Z' be the height of the upper liquid
        surface in the pipe above the datum, and Z be the additional height of the fluid level in the tube
        above the upper liquid surface in the pipe; Z' may be neglected if P1 is measured at the upper
        surface of the liquid in the pipe, or if Z' is small compared with Z.
   v2 = 0 as there is no flow in the tube.
   P2 = 0 if atmospheric pressure is taken as datum and if the top of the tube is open to the atmosphere.
   Z1 = 0 because the datum level is at the mouth of the tube.

The equation then simplifies to

v12/2g + P1/r1 = (Z + Z')g      Z.                                                                          (4.1)

This analysis shows that the differential head on the manometer measures the sum of the velocity head and the pressure head in the flowing liquid.

The Pitot tube can be combined with a piezometer tube, and connected across a common manometer as shown in Fig. 4.1(d). The differential head across the manometer is the velocity head plus the static head of the Pitot tube, less the static head of the piezometer tube. In other words, the differential head measures directly the velocity head of the flowing liquid or gas. This differential arrangement is known as a Pitot-static tube and it is extensively used in the measurement of flow velocities.
We can write for the Pitot-static tube:

             Z   =    v2/2g                                                                                           (4.2)

where Z is the differential head measured in terms of the flowing fluid.


EXAMPLE 4.2. Velocity of air in a duct
Air at 0°C is flowing through a duct in a chilling system. A Pitot-static tube is inserted into the flow line and the differential pressure head, measured in a micromanometer, is 0.8 mm of water. Calculate the velocity of the air in the duct. The density of air at 0°C is 1.3 kg m-3.

From eqn. (4.2) we have

              Z  =   v12/2g

In working with Pitot-static tubes, it is convenient to convert pressure heads into equivalent heads of the flowing fluid, in this case air, using the relationship:
                 
r1Z1   =   r2Z2      from eqn 3.3.

Now 0.8 mm water = 0.8 x 10-3 x
1000
1.3
 
= 0.62 m of air  
 
Also v12   
= 2Zg  
    = 2 x 0.62 x 9.81  

                                      =  12.16 m2s-2

                Therefore   v1   =   3.5 m s-1


Another method of using pressure differentials to measure fluid flow rates is used in Venturi and orifice meters. If flow is constricted, there is a rise in velocity and a fall in static pressure in accordance with Bernouilli's equation. Consider the system shown in Fig. 4.2.

Fig. 4.2. Venturi meter
Figure 4.2. Venturi meter

A gradual constriction has been interposed in a pipe decreasing the area of flow from A1 to A2. If the fluid is assumed to be incompressible and the respective velocities and static pressures are v1 and v2, and P1 and P2, then we can write Bernouilli's equation (eqn.3.7) for the section of horizontal pipe:

           v12/2 + P1/r1 = v22/2+ P2/r2

Furthermore, from the mass balance, eqn. (3.5)

                        A1v1 = A2v2

also, as it is the same fluid

                           r1 = r2 = r

so that we have

            v12/2 + P1/r = (v1A1/A2)2/2 + P2/r

                          v12 = [2(P2 -P1)/r] x A22/(A22 -A12)

By joining the two sections of a pipe to a U-manometer, as shown in Fig. 4.2, the differential head
(P2 -P1)/
r can be measured directly. A manometric fluid of density rm must be introduced, and the head measured is converted to the equivalent head of the fluid flowing by the relationship:

               (P2 -P1)/r = gZrm /r

                     and so Z = (P2 -P1)/rm g

If A1 and A2 are measured, the velocity in the pipe, v1, can be calculated. This device is called a Venturi meter. In actual practice, energy losses do occur in the pipe between the two measuring points and a coefficient C is introduced to allow for this:

v1 = C
[2(P2 -P1 )/r]x A22/(A22 -A12 )

In a properly designed Venturi meter, C lies between 0.95 and 1.0.

The orifice meter operates on the same principle as the Venturi meter, constricting the flow and measuring the corresponding static pressure drop. Instead of a tapered tube, a plate with a hole in the centre is inserted in the pipe to cause the pressure difference. The same equations hold as for the Venturi meter; but in the case of the orifice meter the coefficient, called the orifice discharge coefficient, is smaller. Values are obtained from standard tables, for example British Standard Specification 1042. Orifices have much greater pressure losses than Venturi meters, but they are easier to construct and to insert in pipes.

Various other types of meters are used:
propeller meters where all or part of the flow passes through a propeller, and the rate of rotation of the propeller can be related to the velocity of flow;
impact meters where the velocity of flow is related to the pressure developed on a vane placed in the flow path;
rotameters in which a rotor disc is supported against gravity by the flow in a tapered vertical tube and the rotor disc rises to a height in the tube which depends on the flow velocity.


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Unit Operations in Food Processing. Copyright © 1983, R. L. Earle. :: Published by NZIFST (Inc.)
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