When Venturimeter is inclined, then for a given flow it will show
Same reading
Understanding Venturimeter Reading and Inclination
A Venturimeter is a device used to measure the flow rate of a fluid. It works based on the principle of Bernoulli's equation and the continuity equation. The core idea is that when a fluid flows through a constricted section (the throat) of the Venturimeter, its velocity increases. According to Bernoulli's principle, this increase in kinetic energy is accompanied by a decrease in pressure.
How a Venturimeter Measures Flow Rate
A Venturimeter typically consists of three parts:
- A converging cone: Where the fluid velocity increases and pressure decreases.
- A cylindrical throat: The section with the minimum area and maximum velocity.
- A diverging cone: Where the fluid velocity decreases and pressure recovers (partially).
Pressure taps are located at the inlet (wider section) and the throat (narrower section). The difference in pressure between these two points is measured, usually using a differential manometer or pressure transducers. This measured pressure difference is then used to calculate the flow rate.
Bernoulli's Principle and Inclination
Bernoulli's equation for steady, incompressible, inviscid flow along a streamline states that the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant.
Mathematically, this can be written as:
\[\frac{P}{\rho} + \frac{v^2}{2} + gz = \text{Constant}\]Dividing by \(g\), we get the equation in terms of head:
\[\frac{P}{\rho g} + \frac{v^2}{2g} + z = \text{Constant}\]Where:
- \(P\) is the pressure
- \(\rho\) is the fluid density
- \(v\) is the fluid velocity
- \(g\) is the acceleration due to gravity
- \(z\) is the elevation head (height above a datum)
Applying Bernoulli's equation between the inlet (section 1) and the throat (section 2) of the Venturimeter:
\[\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2\]Rearranging the terms to group pressure and elevation:
\[\left(\frac{P_1}{\rho g} + z_1\right) - \left(\frac{P_2}{\rho g} + z_2\right) = \frac{v_2^2}{2g} - \frac{v_1^2}{2g}\]The term \(\frac{P}{\rho g} + z\) is known as the piezometric head. The difference in piezometric head between section 1 and section 2 is \(\Delta h^* = \left(\frac{P_1}{\rho g} + z_1\right) - \left(\frac{P_2}{\rho g} + z_2\right)\). This is precisely what a differential manometer connected between points 1 and 2 measures.
From the continuity equation, for a given flow rate \(Q\), we have \(Q = A_1 v_1 = A_2 v_2\), where \(A_1\) and \(A_2\) are the areas at the inlet and throat, respectively. Thus, \(v_1 = Q/A_1\) and \(v_2 = Q/A_2\).
Substituting these into the Bernoulli equation:
\[\Delta h^* = \frac{(Q/A_2)^2}{2g} - \frac{(Q/A_1)^2}{2g} = \frac{Q^2}{2g} \left(\frac{1}{A_2^2} - \frac{1}{A_1^2}\right)\] \[\Delta h^* = \frac{Q^2}{2g} \left(\frac{A_1^2 - A_2^2}{A_1^2 A_2^2}\right)\]Solving for \(Q\):
\[Q = \sqrt{2g \Delta h^*} \frac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}}\]This formula shows that the flow rate \(Q\) is directly proportional to the square root of the measured piezometric head difference \(\Delta h^*\). Crucially, the formula for \(Q\) depends only on the geometric properties of the Venturimeter (\(A_1, A_2\)) and the measured piezometric head difference \(\Delta h^*\). The individual elevation terms \(z_1\) and \(z_2\) are accounted for within \(\Delta h^*\).
Conclusion on Venturimeter Inclination
Since the measurement principle relies on the difference in piezometric head, which correctly incorporates any difference in elevation between the measurement points, the orientation (horizontal, vertical, or inclined) of the Venturimeter does not affect the reading for a given flow rate, assuming ideal conditions (steady, incompressible, inviscid flow) and that the device used for pressure measurement correctly reads the piezometric head difference. The Venturimeter is designed to measure the flow rate based on the pressure difference caused by velocity change, and this difference, when correctly measured including hydrostatic effects, remains the same for a given flow rate regardless of inclination.
Therefore, when a Venturimeter is inclined, for a given flow it will show the same reading as it would in a horizontal position.
Important Questions from Fluid Dynamics
The total energy of each particle at various places in the case of perfect incompressible fluid flowing in continuous stream
The coefficient of contraction Cc for an orifice can be determined using other coefficients; discharge and velocity Cv by the relation.
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A pitot static tube is used to measure the velocity of water in a pipe. The stagnation pressure head is 6 m and static pressure head is 5 m. Calculate the velocity of flow assuming the coefficient if tube equal to 0.98.
Navier–stokes equation applies to:
