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Equation Continuity

Reference data and engineering information about equation continuity for fluid mechanics applications.

equationcontinuityCalculator

Overview

The continuity equation expresses conservation of mass in fluid flow. For incompressible flow, the product of cross-sectional area and velocity is constant along a streamtube.

Variables

SymbolDescriptionUnit
AACross-sectional aream²
vvFlow velocitym/s
QQVolume flow ratem³/s
m˙\dot{m}Mass flow ratekg/s

Formula

A1v1=A2v2A_1 v_1 = A_2 v_2

Calculator

Notes

  • Results are approximate and should be verified for critical applications
  • Input values should be within reasonable engineering ranges

Example: Water Flow in a Pipe

Consider a practical application of the Equation of Continuity for incompressible fluids. Water flows through a pipe that narrows from a diameter of 100 mm to 80 mm.

Given:

  • Volumetric flow rate, q=10 m3/hq = 10 \, \text{m}^3/\text{h}
  • Input diameter, d1=100 mm=0.1 md_1 = 100 \, \text{mm} = 0.1 \, \text{m}
  • Output diameter, d2=80 mm=0.08 md_2 = 80 \, \text{mm} = 0.08 \, \text{m}

Step 1: Convert flow rate to consistent SI units q=10 m3h×1 h3600 s=0.00278 m3/sq = 10 \, \frac{\text{m}^3}{\text{h}} \times \frac{1 \, \text{h}}{3600 \, \text{s}} = 0.00278 \, \text{m}^3/\text{s}

Step 2: Calculate cross-sectional areas A1=πd124=π(0.1 m)24=0.00785 m2A_1 = \frac{\pi d_1^2}{4} = \frac{\pi (0.1 \, \text{m})^2}{4} = 0.00785 \, \text{m}^2 A2=πd224=π(0.08 m)24=0.00503 m2A_2 = \frac{\pi d_2^2}{4} = \frac{\pi (0.08 \, \text{m})^2}{4} = 0.00503 \, \text{m}^2

Step 3: Apply the continuity equation for uniform density (Eq. 2) q=v1A1=v2A2q = v_1 A_1 = v_2 A_2

Step 4: Solve for velocities v1=qA1=0.002780.00785=0.354 m/sv_1 = \frac{q}{A_1} = \frac{0.00278}{0.00785} = 0.354 \, \text{m/s} v2=qA2=0.002780.00503=0.553 m/sv_2 = \frac{q}{A_2} = \frac{0.00278}{0.00503} = 0.553 \, \text{m/s}

Interpretation: As the pipe narrows, the fluid velocity increases proportionally to maintain the same mass flow rate, demonstrating the fundamental principle of the Equation of Continuity.

Applications in Engineering

The Equation of Continuity is a foundational principle with broad applications:

  • Piping Systems: Design of water supply, oil/gas transmission, and chemical processing pipelines
  • Aerospace: Airflow analysis in wind tunnels and around aircraft surfaces
  • Environmental Engineering: River flow modeling, flood prediction, and stormwater management
  • Power Generation: Coolant flow in nuclear reactors, steam flow in turbines
  • Biomedical: Blood flow analysis in arteries and veins
  • Microelectronics: Gas flow in semiconductor fabrication processes
  • Industrial Processes: Dairy processing, food production, and hydraulic systems

The principle ensures conservation of mass in steady-state systems and is essential for calculating flow rates, pressures, and velocities in engineering design.

References

Source: engineeringtoolbox.com