What is venturimeter? Derive the expression for measuring rate of flow of fluid in a horizontal pipe.

Introduction to Venturimeter

A venturimeter is a device used for measuring the rate of flow of fluid through a pipe. It is named after its inventor, Giovanni Battista Venturi, an Italian physicist. The venturimeter works on the principle of Bernoulli's theorem, which states that the sum of pressure energy, kinetic energy and potential energy per unit volume is constant at all points along the streamline.

The venturimeter consists of three main components: a converging section, a throat and a diverging section. The converging section reduces the cross-sectional area of the flow, increasing the velocity of the fluid. The throat is the section with the smallest cross-sectional area where the fluid velocity is the highest. The diverging section gradually increases the cross-sectional area, reducing the fluid velocity and increasing its pressure.

Bernoulli's Theorem

Bernoulli's theorem is a fundamental principle in fluid dynamics that describes the conservation of energy for flowing fluids. It states that for an inviscid flow of a nonconducting fluid, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

The mathematical expression of Bernoulli's theorem is:

P1/ρg + V1²/2g + Z1 = P2/ρg + V2²/2g + Z2

where:

• P1 and P2 are the pressures at two points,
• ρ is the fluid density,
• g is the acceleration due to gravity,
• V1 and V2 are the velocities of the fluid at two points,
• Z1 and Z2 are the heights above a reference plane at the two points.

Derivation of the Expression for Measuring Rate of Flow of Fluid

For the derivation, we make the following assumptions: the flow is steady, the fluid is incompressible, there are no frictional losses, and the fluid is flowing horizontally so the potential energy change is zero.

Applying Bernoulli's theorem at the inlet (section 1) and at the throat (section 2), we get:

P1/ρg + V1²/2g = P2/ρg + V2²/2g

The equation for the pressure difference as indicated by the manometer is:

P1 - P2 = ρg * h

where h is the height of the fluid column in the manometer.

Substituting the pressure difference in Bernoulli's equation, we get:

V1²/2g - V2²/2g = h

Solving for V2 (velocity at the throat), we get:

V2 = √(V1² + 2gh)

The rate of flow (Q) is given by the product of the cross-sectional area (A) and the velocity (V). Therefore, at the inlet and the throat, we have:

Q = A1V1 = A2V2

Substituting V2 in the above equation, we get the expression for the rate of flow:

Q = A1A2/√(A1²-A2²) * √(2gh)

where A1 and A2 are the cross-sectional areas at the inlet and the throat, respectively.

Example

Given: Diameter at the inlet (D1) = 0.3 m, Diameter at the throat (D2) = 0.15 m, h = 0.5 m, g = 9.81 m/s².

Required: Rate of flow of fluid.

Solution: First, we calculate the cross-sectional areas at the inlet and the throat:

A1 = π/4 * D1² = π/4 * (0.3)² = 0.071 m² A2 = π/4 * D2² = π/4 * (0.15)² = 0.018 m²

Substituting the given values in the derived expression, we get:

Q = (0.071)(0.018)/√((0.071)²-(0.018)²) * √(2*9.81*0.5) = 0.014 m³/s

Conclusion

In conclusion, a venturimeter is a useful device for measuring the rate of flow of fluid through a pipe. The derived expression for the rate of flow, Q = A1A2/√(A1²-A2²) * √(2gh), is important for practical applications as it allows us to calculate the rate of flow given the dimensions of the venturimeter and the manometer reading.

Diagram

Drawing a diagram of the venturimeter showing the converging section, throat, diverging section, and the manometer would be helpful in understanding the working principle of the venturimeter and the derivation of the expression for the rate of flow.

Summary

A venturimeter is a device used for measuring the rate of flow of fluid through a pipe. It works on the principle of Bernoulli's theorem, which states that the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant at all points along the streamline. The venturimeter consists of a converging section, a throat, and a diverging section. The rate of flow of fluid can be calculated using the derived expression Q = A1A2/√(A1²-A2²) * √(2gh), where A1 and A2 are the cross-sectional areas at the inlet and the throat, respectively.

Analogy

A venturimeter can be compared to a traffic bottleneck. As the road narrows, the cars have to slow down and the traffic flow rate decreases. Similarly, in a venturimeter, as the cross-sectional area decreases, the fluid velocity increases and the flow rate can be measured.