1.7 Conservation of Mass Flow Rate in Fluids

1.7 Conservation of Mass Flow Rate in Fluids

Introduction

The principle of conservation of mass flow rate is a cornerstone of fluid mechanics, ensuring that the mass of a fluid entering a system equals the mass leaving it—assuming no mass is added or lost. This principle, often expressed mathematically through the continuity equation, simplifies analyzing fluid systems.

Flow Rate (f)

Flow rate is defined as the product of the fluid’s velocity and the cross-sectional area through which the fluid flows:

Where:

• : Flow rate (m³/s)

• : Fluid velocity (m/s)

• : Cross-sectional area (m²)

Continuity Equation

The continuity equation ensures that the flow rate remains constant at all points in a system for an incompressible fluid:

Where:

• and : Cross-sectional areas at points 1 and 2

• and : Velocities at points 1 and 2

Note:

• When the cross-sectional area increases, velocity decreases.

• When the cross-sectional area decreases, velocity increases.

Example Problem 1:

Problem: A point in a pipe, , has a radius of meters and a velocity of 20 m/s. Another point, , has a radius of 1.5. Find the velocity at .

Solution:

1. Calculate the ratio of areas:

2. Using the continuity equation:

Relationship Between Area, Velocity, and Pressure

The relationship between these quantities can be summarized as follows:

• Larger Area: Larger pressure, smaller velocity.

• Smaller Area: Smaller pressure, larger velocity.

Key Points

1. Conservation of Mass Flow Rate:

   • Mass flow rate remains constant unless mass is added or removed.

   • Expressed as:

2. Mass Flow Rate Formula:

3. Incompressible Fluids:

   • The continuity equation is most accurate for incompressible fluids like water.

   • Results for gases are approximations.

Example Problem 2:

Problem: A tank contains water at 20°C. Water is pumped out at 2 kg/s, and an equal amount of water at 40°C is pumped in. What is the mass flow rate of water leaving the tank?

Solution: By conservation of mass flow rate:

Answer: The mass flow rate of water leaving the tank is 2 kg/s.

Example Problem 3:

Problem: Tank A and Tank B are connected by a pipe. Water flows from Tank A to Tank B at 2 kg/s. Simultaneously, 3 kg/s of water is pumped out of Tank B, and 3 kg/s is pumped into Tank B from a separate source. What is the flow rate from Tank A to Tank B?

Solution: Using conservation of mass flow rate:

• Total inflow to Tank B:

• Outflow from Tank B:

The flow rate from Tank A to Tank B remains 2 kg/s.

The conservation of mass flow rate is fundamental in fluid mechanics, bridging concepts of velocity, pressure, and area. This principle, paired with Bernoulli’s equation, equips you with powerful tools to analyze real-world fluid systems. 🌊