Mass flow rate is a measure of the mass of a substance passing through a given surface per unit of time. It is a fundamental concept in fluid dynamics, thermodynamics, and various engineering applications. Mass flow rate is crucial in analyzing and designing systems involving fluid flow, such as pipelines, HVAC systems, and chemical processing plants.

The formula for mass flow rate is:

\[ \dot{m} = \rho \times v \times A \]

Where:

- \( \dot{m} \) is the mass flow rate (in kg/s)
- \( \rho \) is the density of the fluid (in kg/m³)
- \( v \) is the velocity of the fluid (in m/s)
- \( A \) is the cross-sectional area of the flow (in m²)

Let's calculate the mass flow rate for a fluid flowing through a pipe:

- Given:
- Density (\( \rho \)) = 1000 kg/m³ (water at room temperature)
- Velocity (\( v \)) = 2 m/s
- Cross-sectional area (\( A \)) = 0.05 m² (pipe with radius ≈ 12.6 cm)

- Apply the mass flow rate formula: \[ \dot{m} = \rho \times v \times A \]
- Substitute the known values: \[ \dot{m} = 1000 \text{ kg/m³} \times 2 \text{ m/s} \times 0.05 \text{ m²} \]
- Perform the calculation: \[ \dot{m} = 100 \text{ kg/s} \]

Let's visualize the mass flow rate in a pipe:

This diagram illustrates:

- The pipe cross-section (blue outline)
- The fluid density (\( \rho \)) represented by green wavy lines
- The velocity (\( v \)) shown by the blue arrow
- The cross-sectional area (\( A \)) indicated by the yellow circle
- The resulting mass flow rate (\( \dot{m} \)) shown at the bottom

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