How to Calculate Flow from Pressure

Hagen-Poiseuille Equation:

\[ Q = \frac{\pi r^4 \Delta P}{8 \mu L} \]

Pipe Radius (r):

Pressure Difference (ΔP):

Dynamic Viscosity (μ):

Pa·s

Pipe Length (L):

Volumetric Flow Rate (Q):

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1. What is the Hagen-Poiseuille Equation?

Definition: The Hagen-Poiseuille equation describes the volumetric flow rate of a fluid through a cylindrical pipe under laminar flow conditions.

Purpose: It helps engineers and physicists calculate fluid flow rates in pipes, capillaries, and other cylindrical systems.

2. How Does the Equation Work?

The equation is:

\[ Q = \frac{\pi r^4 \Delta P}{8 \mu L} \]

Where:

\( Q \) — Volumetric flow rate (m³/s)
\( r \) — Pipe radius (m)
\( \Delta P \) — Pressure difference (Pa)
\( \mu \) — Dynamic viscosity (Pa·s)
\( L \) — Pipe length (m)

Explanation: The flow rate is directly proportional to the pressure difference and the fourth power of the radius, and inversely proportional to viscosity and pipe length.

3. Importance of Flow Rate Calculation

Details: Accurate flow rate calculations are essential for designing plumbing systems, medical devices (like IVs), industrial processes, and understanding blood flow in arteries.

4. Using the Calculator

Tips:

Enter all values in SI units (meters, pascals, etc.)
Default viscosity is for water at 20°C (0.001002 Pa·s)
Radius has the most significant effect (r⁴ relationship)
All values must be greater than zero

5. Frequently Asked Questions (FAQ)

Q1: What are the limitations of this equation?
A: It only applies to laminar flow (Re < 2000), Newtonian fluids, and straight pipes with constant circular cross-sections.

Q2: How does temperature affect the calculation?
A: Temperature mainly affects viscosity. For water, viscosity decreases by about 2% per °C increase.

Q3: Why is radius to the fourth power so important?
A: This strong dependence means small changes in pipe diameter dramatically affect flow rate (doubling radius increases flow 16x).

Q4: Can I use this for blood flow calculations?
A: It can approximate blood flow in larger vessels, but blood's non-Newtonian properties make it less accurate for small vessels.

Q5: What's a typical flow rate for household plumbing?
A: A ½" pipe (0.00635m radius) with 100kPa pressure might have ~0.0005 m³/s (0.5 L/s) flow.