A horizontal venturimeter in a water main has a 20 cm diameter at one end and tapers to 10 cm at its throat. A piezometer installed at the inlet reads 30 cm, while the one at the throat reads 18 cm. Determine the discharge through the main if Cd = 0.98.

Fluid Mechanics Problem Solution

Problem Statement

A horizontal venturimeter in a water main has a 20 cm diameter at one end and tapers to 10 cm at its throat. A piezometer installed at the inlet reads 30 cm, while the one at the throat reads 18 cm. Determine the discharge through the main if C_d = 0.98.

Given Data

Diameter at inlet (d₁) 20 cm = 0.2 m

Diameter at throat (d₂) 10 cm = 0.1 m

Cross-sectional area of inlet (A₁) π/4 × (0.2)² ≈ 0.0314 m²

Cross-sectional area of throat (A₂) π/4 × (0.1)² ≈ 0.00785 m²

Piezometer at inlet 30 cm = 0.3 m

Piezometer at throat 18 cm = 0.18 m

Head (h) 0.3 m – 0.18 m = 0.12 m

Coefficient of Discharge (C_d) 0.98

Solution Approach

To determine the discharge through the venturimeter, we’ll use the discharge equation derived from Bernoulli’s principle and continuity equation, considering the coefficient of discharge.

Calculations

Discharge Equation Derivation

Step 1: The discharge (Q) through the venturimeter is given by:

Q = C_d √(2g h) (A₁ A₂) / √(A₁² – A₂²)

This equation is derived from the Bernoulli equation applied between the inlet and throat sections, combined with the continuity equation.

Step 2: Substituting the provided values (with g = 9.81 m/s²):

Q = 0.98 √(2 × 9.81 × 0.12) × (0.0314 × 0.00785) / √((0.0314)² – (0.00785)²)

Step 3: Simplifying the calculation:

Q = 0.98 × √(2.354) × 0.000246 / √(0.000985 – 0.000062)

Q = 0.98 × 1.534 × 0.000246 / √(0.000923)

Q = 0.98 × 1.534 × 0.000246 / 0.0304

Q = 0.98 × 1.534 × 0.0081 ≈ 0.0122 m³/s

Discharge (Q) = 0.0122 m³/s = 12.2 liters/s

Detailed Explanation

Working Principle of Venturimeter

A venturimeter works on the principle of Bernoulli’s theorem, which states that in a flowing fluid, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline.

Pressure-Velocity Relationship

As the fluid flows from the wider section (inlet) to the narrower section (throat), its velocity increases due to the continuity equation (A₁V₁ = A₂V₂). According to Bernoulli’s principle, this increase in velocity leads to a decrease in pressure at the throat compared to the inlet. This pressure difference is directly related to the flow rate.

Role of Coefficient of Discharge

The coefficient of discharge (C_d) accounts for energy losses due to friction and the non-ideal flow pattern. A value of 0.98 indicates that the venturimeter is operating at 98% efficiency compared to an ideal, frictionless venturimeter.

Practical Applications

Venturimeters are widely used in industrial applications for measuring flow rates of liquids and gases in pipes. They are particularly valuable in:

Water supply systems for monitoring flow rates
Industrial processes where accurate flow measurement is critical
Laboratories for precise fluid flow experiments
Irrigation systems for water management

Advantages of Venturimeters

Compared to other flow measurement devices, venturimeters offer:

High accuracy and reliability
Lower permanent pressure loss (higher energy efficiency)
Simple construction with no moving parts
Durable and long-lasting performance
Suitability for a wide range of fluids including those with suspended solids

Analysis of Results

The calculated discharge of 0.0122 m³/s (12.2 liters/s) represents the volumetric flow rate through the water main. This is a moderate flow rate typical for medium-sized water distribution systems. The accuracy of this calculation depends on several factors:

The precision of the piezometer readings
The accuracy of the venturimeter calibration (reflected in the C_d value)
The validity of assumptions regarding steady, incompressible flow

For engineers and students, understanding the relationship between pressure differences and flow rates is fundamental in fluid mechanics and has widespread applications in hydraulic engineering, environmental engineering, and mechanical systems design.