Flow Rate Through Pipe
Flow rate through a pipe is usually measured by providing a coaxial area contraction within the pipe and by recording the pressure drop across the contraction. Therefore the determination of the flow rate from the measurement of pressure drop depends on the straight forward application of Bernoulli’s equation.
Three different flow meters operate on this principle.
• Venturi meter
• Orifice meter
• Flow nozzle.
Venturimeter
Construction: A Venturimeter is essentially a short pipe (Fig. 1) consisting of two conical parts with a short portion of uniform cross-section in between. This short portion has the minimum area and is known as the throat. The two conical portions have the same base diameter, but one is having a shorter length with a larger cone angle while the other is having a larger length with a smaller cone angle.
Working:
• The Venturimeter is always used in a way that the upstream part of the flow takes place through the short conical portion while the downstream part of the flow through the long one.
• This ensures a rapid converging passage and a gradual diverging passage in the direction of flow to avoid the loss of energy due to separation. In course of a flow through the converging part, the velocity increases in the direction of flow according to the principle of continuity, while the pressure decreases according to Bernoulli’s theorem.
• The velocity reaches its maximum value and pressure reaches its minimum value at the throat. Subsequently, a decrease in the velocity and an increase in the pressure takes place in course of flow through the divergent part. This typical variation of fluid velocity and pressure by allowing it to flow through such a constricted convergent-divergent passage was first demonstrated by an Italian scientist Giovanni Battista Venturi in 1797.
Figure 2 shows that a Venturimeter is inserted in an inclined pipe line in a vertical plane to measure the flow rate through the pipe. Let us consider a steady, ideal and one dimensional (along the axis of the venturi meter) flow of fluid. Under this situation, the velocity and pressure at any section will be uniform.
Let the velocity and pressure at the inlet (Sec. 1) are V1 and p1 respectively, while those at the throat (Sec. 2) are V2 and p2. Now, applying Bernoulli’s equation between Secs 1 and 2, we get
where ρ is the density of fluid flowing through the Venturimeter.
From continuity,
V1A1 = V2A2 (3)
where A1 and A2 are the cross-sectional areas of the venturi meter at its throat and inlet respectively.
With the help of Eq. (3), Eq. (2) can be written as
Where h1* and h2* are the piezometric pressure heads at sec. 1 and sec. 2 respectively, and are defined as
Hence, the volume flow rate through the pipe is given by
If the pressure difference between Sections 1 and 2 is measured by a manometer as shown in Fig. 2, we can write
where ρm is the density of the manometric liquid.
Equation (7) shows that a manometer always registers a direct reading of the difference in piezometric pressures. Now, substitution of h1* – h2* from Eq. (15.7) in Eq. (15.6) gives
• If the pipe along with the Venturimeter is horizontal, then z1 = z2; and hence h1* – h2* becomes h1 − h2, where h1 and h2 are the static pressure heads 
The manometric equation Eq. (7) then becomes
Therefore, it is interesting to note that the final expression of flow rate, given by Eq. (8), in terms of manometer deflection ∆h, remains the same irrespective of whether the pipe-line along with the venturimeter connection is horizontal or not.
Measured values of ∆h, the difference in piezometric pressures between Secs I and 2, for a real fluid will always be greater than that assumed in case of an ideal fluid because of frictional losses in addition to the change in momentum.
Therefore, Eq. (8) always overestimates the actual flow rate. In order to take this into account, a multiplying factor Cd, called the coefficient of discharge, is incorporated in the Eq. (8) as
The coefficient of discharge Cd is always less than unity and is defined as
where, the theoretical discharge rate is predicted by the Eq. (8) with the measured value of ∆h, and the actual rate of discharge is the discharge rate measured in practice.
Value of Cd for a Venturimeter usually lies between 0.95 to 0.98.
Orifice meter
Construction: An orifice meter provides a simpler and cheaper arrangement for the measurement of fow through a pipe. An orifice meter is essentially a thin circular plate with a sharp edged concentric circular hole in it.
Working:
• The orifice plate, being fixed at a section of the pipe, (Fig. 3) creates an obstruction to the flow by providing an opening in the form of an orifice to the flow passage.
• The area A0 of the orifice is much smaller than the cross-sectional area of the pipe. The flow from an upstream section, where it is uniform, adjusts itself in such a way that it contracts until a section downstream the orifice plate is reached, where the vena contracta is formed, and then expands to fill the passage of the pipe.
• One of the pressure tapings is usually provided at a distance of one diameter upstream the orifice plate where the flow is almost uniform (Sec. 1-1) and the other at a distance of half a diameter downstream the orifice plate.
• Considering the fluid to be ideal and the downstream pressure taping to be at the vena contracta (Sec. c-c), we can write, by applying Bernoulli’s theorem between Sec. 1-1 and Sec. c-c,
Where p1* and pc* are the piezometric pressures at Sec.1-1 and c-c respectively.
From the equation of continuity,
V1A1 = VcAc (11)
where Ac is the area of the vena contracta.
With the help of Eq. (11), Eq. (10) can be written as,
Flow Nozzle
• The flow nozzle as shown in Fig. 4 is essentially a venturi meter with the divergent part omitted. Therefore the basic equations for calculation of flow rate are the same as those for a venturimeter.
• The dissipation of energy downstream of the throat due to flow separation is greater than that for a Venturimeter. But this disadvantage is often offset by the lower cost of the nozzle.
• The downstream connection of the manometer may not necessarily be at the throat of the nozzle or at a point sufficiently far from the nozzle.
• The deviations are taken care of in the values of Cd, The coefficient Cd depends on the shape of the nozzle, the ratio of pipe to nozzle diameter and the Reynolds number of flow.
A comparative picture of the typical values of Cd, accuracy, and the cost of three flow meters (Venturimeter, orifice meter and flow nozzle) is given below:
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