Venturi meter is a flow measurement instrument or device used to measure discharge through a pipe. It is based on Bernoulli’s principle.
Construction of Venturi Meter:
A venturi meter is
essentially a short pipe, Fig.1, consisting of two conical parts with a short
portion of the 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.
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| Fig.1: Venturi Meter (I) Schematic (II) Real One Used In Practice |
Working of Venturi Meter:
The variation in the conical portion upstream and downstream ensures a rapid converging passage and a gradual diverging passage in the direction of flow. This is just to avoid the loss of energy due to separation at the throat. During 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 and pressure reaches its minimum at the throat. Subsequently, a decrease in the velocity and an increase in the pressure take place in course of flow through the divergent part. Fig.2. shows that a venturi meter is inserted in an inclined pipeline in a vertical plane to measure the flow rate through the pipe. Let us consider a steady, ideal, and one-dimensional (along with the axis of the venturi meter) flow of fluid. Under this situation, the velocity and pressure at any section will be uniform.
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| Fig.2: Measurement of Flow by a Venturimeter |
Let the velocity and
pressure at the inlet (a) be V1 and P1, respectively,
while those at the throat (b) are V2 and P2. Now,
applying Bernoulli’s equation between (a) and (b), we get
`\frac{P_1}{\rho g}+\frac{V_1^2}{2g}+z_1=\frac{P_2}{\rho g}+\frac{V_2^2}{2g}+z_2 ...(1)`
where ρ is the density of
the fluid flowing through the venturi meter. From continuity,
`V_1A_1=V_2A_2 ...(2)`
where A1 and A2
are the cross-sectional areas of the venturi meter at its throat and inlet
respectively.
`+\frac{V_2^2}{2g}\left(1-\frac{A_2^2}{A_1^2}\right)=\left(\frac{P_1}{2g}+z_1\right)-\left(\frac{P_2}{2g}+z_2\right) ...(3)`
With the help of equation
(1), equation (2) can be written as
`V_2=\frac1{\sqrt{1-\frac{A_2^2}{A_1^2}}}\sqrt{2g(h_1^\ast-h_2^\ast)} ...(4)`
where, h1*
and h2* are the
piezometric pressure heads at (a) and (b), respectively, and are defined as
`h_1^\ast=\frac{P_2}{\rho g}+z_1 ...(5)`
`h_2^\ast=\frac{P_2}{\rho g}+z_2 ...(6)`
Hence, the volume flow
rate through the pipe is given by
`Q=A_2V_2=\frac1{\sqrt{1-\frac{A_2^2}{A_1^2}}}\sqrt{2g(h_1^\ast}-h_2^\ast) ...(7)`
If the pressure
difference between (a) and (b) is measured by a manometer as shown in Fig.1, we
can write
`P_1+\rho g(z_1-h_0)=P_2+\rho g(z_2-h_0-\triangle h)+\h\rho_mg ...(8)`
`(P_1+\rho gz_1)-(P_2+\rho gz_2)=(\rho_m-\rho) g\triangle h ...(9)`
`\left(\frac{P_1}{\rho g}+z_1\right)-\left(\frac{P_2}{\rho g}+z_2\right)=\left(\frac{\rho_m}\rho-1\right)\triangle h ...(10)`
`h_1^\ast-h_2^\ast=\left(\frac{\rho_m}\rho-1\right)\triangle h ...(11)`
where ρm is
the density of the manometric liquid. Equation (11) shows that a manometer
always registers a direct reading of the difference in piezometric pressures.
Now, the substitution of h1* − h2* from
equation (11) in equation (10) gives,
`Q=\frac{A_1A_2}{\sqrt{A_1^2}A_2^2}\sqrt{2g\left(\frac{\rho_m}\rho-1\right)}\triangle h ...(12)`
If the pipe along with
the venturi meter is horizontal, then Z1 = Z2; and hence
h1* − h2*
becomes h1 − h2,
where h1 and h2 are the static pressure heads.
`h_1=\frac{P_1}{\rho g} ...(13)`
`h_2=\frac{P_2}{\rho g} ...(14)`
Thus, manometric equation
(11) becomes,
`h_1-h_2=\left(\frac{\rho_m}\rho-1\right)\triangle h ...(15)`
The final expression of
flow rate given by equation (12), in terms of manometer deflection ∆h, remains
the same irrespective of whether the pipeline along with the venturi meter
connection is horizontal or not. The measured values of ∆h, the difference in
piezometric pressures between (a) and (b), for a real fluid will always be
greater than that of an ideal fluid because of frictional losses in addition to
the change in momentum. Therefore, equation (12) always overestimates the
actual flow rate. To compensate for this a multiplying factor (Cd),
called the coefficient of discharge, is incorporated in equation (12) as
`Q_{Actual}=C_d\frac{A_1A_2}{\sqrt{A_1^2}-A_2^2}\sqrt{2g\left(\frac{\rho_m}\rho-1\right)\triangle h} ...(16)`
The coefficient of
discharge Cd is always less than unity and is defined as;
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| Equation (17) |
Where the theoretical discharge rate is predicted by Eq. (12) with the measured value of ∆h, and the actual rate of discharge is the discharge rate measured in practice. The value of Cd for a venturi meter usually lies between 0.95 to 0.98.
Applications of Venturi Meter:
- Venturimeter can be used for the measurement of the flow of water, liquids, gases, dirty liquids, etc.
- They are commonly used in the water supply industry.
Advantages of Venturi Meter:
- It has a low head loss of about 10% of differential pressure head.
- It can measure higher flow rates in pipes having few meters of diameters due to the high coefficient of discharge owing to lower loss.
- It is suitable for use in any position, for example, horizontal, vertical, or inclined.
- Higher sensitivities can be achieved due to smaller size throat which leads to higher pressure differential.
Disadvantages of Venturi Meter:
- It has space limitations due to its larger size.
- Due to its large size, the cost of a venturi meter is higher.
- The very small diameter of the throat results in cavitations of fluid in the throat.
- It is more susceptible to errors due to burrs or deposits around the downstream (throat) tapping.
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