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'

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  =   v^₁2/2g

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.

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 r_m must be introduced, and the head measured is converted to the equivalent head of the fluid flowing by the relationship:

               (P₂ -P₁)/r = gZr_m /r

                     and so Z = (P₂ -P₁)/r_m 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.

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