Continuity Equation Calculator
Calculate continuity equation calculator calculate exit velocity v2 or area A2 in pipe flow constrictions using the Continuity Equation (A1 * v1 = A2 * v2). Accurate physics formulas and unit conversions for engineers, students & technicians.
Calculate Pipe Constriction Velocity (A1 · v_1 = A2 · v_2)
Enter your physical parameters below to compute verified fluid velocity metrics.
Calculation Results
Calculated using verified physical methodology: Continuity Equation: A1 \cdot v_1 = A2 \cdot v_2
Exit Velocity: v_2 = v_1 \left(\A1 / A2\right)
Quick Summary
The Continuity Equation Calculator evaluates fluid conservation of mass ($A1 \cdot v_1 = A2 \cdot v_2$) in pipe constrictions and nozzles, computing exit velocity ($v_2 = v_1 \A1 / A2$) in m/s, km/h, ft/s, and velocity speedup ratios.
Formula Explanation
Continuity Equation: A1 \cdot v_1 = A2 \cdot v_2
Exit Velocity: v_2 = v_1 \left(\A1 / A2\right)
How It Works
The Continuity Equation Calculator multiplies entry pipe area ($A1$) by entry fluid velocity ($v_1$) to find conserved flow rate ($Q$). It divides flow rate ($Q$) by exit area ($A2$) to output exit velocity ($v_2$), speedup ratio ($A1 / A2$), and flow rate in L/min.
Step-by-Step Worked Example
Practical Problem: Water enters a pipe reducer nozzle with area $A1 = 0.05\text{ m}^2$ at velocity $v_1 = 2.0\text{ m/s}$. The nozzle area constricts to $A2 = 0.01\text{ m}^2$. Calculate exit fluid velocity $v_2$.
- Step 1: Identify Input Parameters: $A1 = 0.05\text{ m}^2$, $v_1 = 2.0\text{ m/s}$, $A2 = 0.01\text{ m}^2$.
- Step 2: Calculate Conserved Flow Rate ($Q = A1 \cdot v_1$): $Q = 0.05 \times 2.0 = 0.100\text{ m}^3/\text{s} = 6,000.0\text{ L/min}$.
- Step 3: Calculate Area Amplification Ratio ($A1 / A2$): $\text{Ratio} = \0.05 / 0.01 = 5.00\text{ (5x area constriction)}$.
- Step 4: Apply the Continuity Equation for Exit Velocity ($v_2$): $v_2 = v_1 \times 5.00 = 2.0\text{ m/s} \times 5.00 = 10.00\text{ m/s}$.
- Step 5: Format Final Velocity Metrics: Exit Velocity $v_2 = 10.00\text{ m/s} = 36.00\text{ km/h} = 32.81\text{ ft/s}$. Conserved Flow Rate = $6,000.0\text{ L/min}$. Speedup Multiplier = $5.0x$.
Real-World Calculation Examples
Scenario 1: Industrial Pipe Reducer Nozzle
Parameters: $A1 = 0.05\text{ m}^2$, $v_1 = 2.0\text{ m/s}$, $A2 = 0.01\text{ m}^2$
Result: $v_2 = 10.00\text{ m/s}$ (36.0 km/h, 5.0x speedup). Pipe constriction velocity.
Scenario 2: Garden Hose Finger Pinch Squeeze
Parameters: $A1 = 3.14\text{ cm}^2$, $v_1 = 1.5\text{ m/s}$, $A2 = 0.785\text{ cm}^2$ (75% pinch)
Result: $v_2 = 6.00\text{ m/s}$ (21.6 km/h, 4.0x speedup). Hose nozzle jet spray.
Scenario 3: Fire Engine Hose Taper Nozzle
Parameters: $A1 = 50\text{ cm}^2$, $v_1 = 3.0\text{ m/s}$, $A2 = 5\text{ cm}^2$ (90% reduction)
Result: $v_2 = 30.00\text{ m/s}$ (108.0 km/h, 10.0x speedup). High-speed fire jet spray.
Scenario 4: River Channel Narrowing Canyon Stream
Parameters: $A1 = 100\text{ m}^2$, $v_1 = 1.0\text{ m/s}$, $A2 = 25\text{ m}^2$
Result: $v_2 = 4.00\text{ m/s}$ (14.4 km/h, 4.0x speedup). River rapid canyon acceleration.
Key Benefits of Using This Calculator
Conservation of Mass Verification
Verifies incompressible fluid mass flow rate conservation across varying cross-sections ($A1 v_1 = A2 v_2$).
Nozzle & Reducer Pipe Design
Sizes nozzle tip diameters and pipe reducer fittings for fire hoses, sprayers, and jet engines.
Multi-Unit Readouts
Outputs velocity in m/s, km/h, and Imperial ft/s.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is the Continuity Equation in fluid dynamics?
The Continuity Equation expresses the Law of Conservation of Mass for fluid flow: mass entering a pipe per unit time must equal mass exiting (rho1 * A1 * v1 = rho2 * A2 * v2).
What is the Continuity Equation formula for incompressible fluids?
For incompressible fluids (constant density rho), area and velocity product is constant: A1 * v1 = A2 * v2 (or Q1 = Q2).
Why does water speed up when squeezing a hose tip?
Pinch-squeezing the tip reduces exit area A2; since total flow rate Q = A * v is constant, decreasing area A forces fluid velocity v to increase dramatically.
How does pipe diameter change affect fluid velocity?
Since area A = pi * (d/2)², velocity v is inversely proportional to diameter squared (v2 / v1 = (d1 / d2)²); halving pipe diameter quadruples fluid velocity.
Does the Continuity Equation apply to gases?
For compressible gases at high speeds, density changes must be included: rho1 * A1 * v1 = rho2 * A2 * v2.
How relates Continuity Equation to Bernoulli's Equation?
The Continuity Equation determines velocity change v2 from area change A2; Bernoulli's Equation then uses this new velocity v2 to calculate static pressure P2.
What is volumetric flow rate Q?
Q = A1 * v1 = A2 * v2 (constant volume discharge in m³/s or L/min).
What is 1D steady-state flow assumption?
Assumes uniform average velocity across pipe cross-sections and constant mass flow rate over time.
How converts m/s to km/h?
Multiply m/s by 3.6 to obtain km/h (e.g. 10 m/s = 36 km/h).
How converts m/s to ft/s?
Multiply m/s by 3.28084 to obtain feet per second (ft/s) (e.g. 10 m/s = 32.81 ft/s).