A 30 cm diameter pipe carries oil of sp. gr. 0.8 at a velocity of 2m/s. At another section the diameter is 20cm. Compute the velocity at this section and discharge in m3/s and kg/s.

Problem Statement

A 30 cm diameter pipe carries oil of specific gravity 0.8 at a velocity of 2 m/s. At another section, the diameter is 20 cm. Compute the velocity at this section and discharge in m³/s and kg/s.

Solution

1. Given:

   Diameter at section 1: \( d_1 = 30 \) cm = 0.3 m
   Area at section 1: \( A_1 = \frac{\pi}{4} (0.3)^2 = 0.07068 \) m²
   Velocity at section 1: \( V_1 = 2 \) m/s
   Diameter at section 2: \( d_2 = 20 \) cm = 0.2 m
   Area at section 2: \( A_2 = \frac{\pi}{4} (0.2)^2 = 0.0314 \) m²
   

2. Applying Continuity Equation:

   Discharge at section 1:

   \( Q_1 = A_1 V_1 = 0.07068 \times 2 = 0.1413 \) m³/s

   Since \( Q_1 = Q_2 \), we have:

   \( Q_2 = 0.1413 \) m³/s

3. Computing Velocity at Section 2:

   \( V_2 = \frac{Q_2}{A_2} = \frac{0.1413}{0.0314} = 4.5 \) m/s

4. Expressing Q in kg/s:

   \( Q_2 = \text{Density} \times Q_2 = (0.8 \times 1000) \times 0.1413 = 113 \) kg/s

Physical Significance & Engineering Context

Fundamental Concepts

• Volumetric Flow Rate (Q): Measures the volume of fluid passing through a cross-section per unit time. Fundamental in pipe system design, irrigation planning, and hydraulic engineering.
• Cross-sectional Area (A): Determines the flow capacity. The pipe diameter exponentially affects flow rate since area ∝ diameter².
• Flow Velocity (V): Represents the average speed of fluid particles. Depends on pressure gradient and pipe roughness.

Practical Interpretation

• The calculated 113 kg/s means 113 kg of oil flows through every second.
• If pipe diameter is doubled, flow rate would increase fourfold at the same velocity.
• Used in oil transport, water supply systems, and industrial applications.

Measurement Techniques

• Venturi meters (pressure difference measurement).
• Ultrasonic flow sensors.
• Rotational turbine meters.
• Volumetric tank measurements.