A horizontal venturimeter with inlet and throat diameters of 400mm and 200mm respectively, is connected to a pipe. If the pressure in the inlet portion is 200KPa and the vacuum pressure (negative) on the throat is 400mm of mercury, find the rate of flow in the pipe taking Cd = 0.97.

Horizontal Venturimeter Flow Measurement

Problem Statement

A horizontal venturimeter with an inlet diameter of 400 mm and a throat diameter of 200 mm is connected to a pipe carrying water. The pressure in the inlet section is 200 kPa while the throat experiences a vacuum pressure corresponding to -400 mm of mercury. Given a discharge coefficient (C_d) of 0.97, determine the rate of flow in the pipe using the continuity and energy equations.

Given Data

Inlet Diameter (d₁)	400 mm = 0.4 m
Inlet Area (A₁)	(π/4) × (0.4)² ≈ 0.1256 m²
Throat Diameter (d₂)	200 mm = 0.2 m
Throat Area (A₂)	(π/4) × (0.2)² ≈ 0.0314 m²
Pressure at Inlet (P₁)	200 kPa = 200×10³ N/m²
Throat Pressure Head	-400 mm Hg
Discharge Coefficient (C_d)	0.97
Density of Water (ρ)	1000 kg/m³
Acceleration due to Gravity (g)	9.81 m/s²
Conversion: 1 m Hg	≈ 13.6 m of water

1. Calculating Pressure Heads

The pressure head at the inlet is:

P₁/ρg = 200000/(1000×9.81) ≈ 20.38 m of water

The throat pressure head, given the vacuum of -400 mm Hg, is converted as:

P₂/ρg = -0.4 m Hg = -0.4×13.6 = -5.44 m of water

Since the pipe is horizontal, Z₁ = Z₂. The net effective head difference is:

h = (P₁ - P₂)/(ρg) = 20.38 - (-5.44) = 25.82 m

2. Energy Equation and Flow Rate

The discharge through the venturimeter is given by:

Q = C_d × √(2gh) × (A₁ A₂)/√(A₁² - A₂²)

Substituting the values:

Q = 0.97 × √(2×9.81×25.82) × (0.1256×0.0314)/√(0.1256² - 0.0314²)

Evaluating the expression gives:

Q ≈ 0.708 m³/s

Conclusion

By applying the continuity and energy equations with the given pressure heads, the flow rate through the horizontal venturimeter is determined to be approximately 0.708 m³/s.