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Poiseuille's Law Formula Calculator

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

1. What is the Poiseuille's Law Formula Calculator?

Definition: This calculator computes the volumetric flow rate (\(Q\)) of a fluid through a cylindrical pipe using Poiseuille's Law, given by \( Q = \frac{\pi r^4 \Delta P}{8 \eta L} \), where \(r\) is the radius of the pipe, \(\Delta P\) is the pressure difference, \(\eta\) is the viscosity of the fluid, and \(L\) is the length of the pipe.

Purpose: It is used in fluid dynamics to determine the flow rate of a Newtonian fluid under laminar flow conditions, applicable in medical science (e.g., blood flow in vessels), engineering (e.g., pipe flow), and physics.

2. How Does the Calculator Work?

The calculator uses Poiseuille's Law:

Formula: \[ Q = \frac{\pi r^4 \Delta P}{8 \eta L} \] where:

  • \(Q\): Flow rate (m³/s, L/s, ft³/s)
  • \(r\): Radius (m, cm, ft, in)
  • \(\Delta P\): Pressure difference (Pa, kPa, atm)
  • \(\eta\): Viscosity (Pa·s, cP)
  • \(L\): Length (m, cm, ft, in)

Unit Conversions:

  • Radius and Length:
    • 1 m = 1 m
    • 1 cm = 0.01 m
    • 1 ft = 0.3048 m
    • 1 in = 0.0254 m
  • Pressure Difference:
    • 1 Pa = 1 Pa
    • 1 kPa = 1000 Pa
    • 1 atm = 101325 Pa
  • Viscosity:
    • 1 Pa·s = 1 Pa·s
    • 1 cP = 0.001 Pa·s
  • Flow Rate (Output):
    • 1 m³/s = 1 m³/s
    • 1 L/s = 0.001 m³/s
    • 1 ft³/s = 0.028316846592 m³/s
The flow rate is calculated in m³/s and can be converted to the selected output unit (m³/s, L/s, ft³/s). Results greater than 10,000 or less than 0.001 are displayed in scientific notation; otherwise, they are shown with 4 decimal places.

Steps:

  • Enter the radius (\(r\)), pressure difference (\(\Delta P\)), viscosity (\(\eta\)), and length (\(L\)) with their units (default: \(r = 0.01 \, \text{m}\), \(\Delta P = 1000 \, \text{Pa}\), \(\eta = 0.001 \, \text{Pa·s}\), \(L = 1 \, \text{m}\)).
  • Convert inputs to SI units (m, Pa, Pa·s, m).
  • Validate that radius, viscosity, and length are greater than 0, and pressure difference is non-negative.
  • Calculate the flow rate in m³/s using the formula.
  • Convert the flow rate to the selected output unit.
  • Display the result, using scientific notation if the value is greater than 10,000 or less than 0.001, otherwise rounded to 4 decimal places.

3. Importance of Poiseuille's Law Calculation

Calculating the flow rate using Poiseuille's Law is crucial for:

  • Fluid Dynamics: Understanding the flow of fluids through pipes or tubes under laminar conditions, such as water in pipelines or air in ventilation systems.
  • Medical Science: Analyzing blood flow in capillaries, where Poiseuille's Law helps model flow rates and resistance in the circulatory system.
  • Engineering: Designing systems like hydraulic circuits, microfluidic devices, and pipelines, where flow rate determines performance and efficiency.

4. Using the Calculator

Examples:

  • Example 1: Calculate the flow rate for \(r = 0.01 \, \text{m}\), \(\Delta P = 1000 \, \text{Pa}\), \(\eta = 0.001 \, \text{Pa·s}\), \(L = 1 \, \text{m}\), output in m³/s:
    • Enter \(r = 0.01 \, \text{m}\), \(\Delta P = 1000 \, \text{Pa}\), \(\eta = 0.001 \, \text{Pa·s}\), \(L = 1 \, \text{m}\).
    • Radius term: \(r^4 = (0.01)^4 = 0.00000001 \, \text{m}^4\).
    • Numerator: \(\pi r^4 \Delta P = \pi \times 0.00000001 \times 1000 \approx 3.14159 \times 10^{-5} \, \text{m}^4 \cdot \text{Pa}\).
    • Denominator: \(8 \eta L = 8 \times 0.001 \times 1 = 0.008 \, \text{Pa·s·m}\).
    • Flow rate: \(Q = \frac{3.14159 \times 10^{-5}}{0.008} \approx 0.003927 \, \text{m}^3/\text{s}\).
    • Output unit: m³/s (no conversion needed).
    • Result: \( \text{Flow Rate} = 0.0039 \, \text{m}^3/\text{s} \).
  • Example 2: Calculate the flow rate for \(r = 1 \, \text{cm}\), \(\Delta P = 1 \, \text{kPa}\), \(\eta = 1 \, \text{cP}\), \(L = 39.37008 \, \text{in}\), output in L/s:
    • Enter \(r = 1 \, \text{cm}\), \(\Delta P = 1 \, \text{kPa}\), \(\eta = 1 \, \text{cP}\), \(L = 39.37008 \, \text{in}\).
    • Convert: \(r = 1 \times 0.01 = 0.01 \, \text{m}\), \(\Delta P = 1 \times 1000 = 1000 \, \text{Pa}\), \(\eta = 1 \times 0.001 = 0.001 \, \text{Pa·s}\), \(L = 39.37008 \times 0.0254 = 1 \, \text{m}\).
    • Radius term: \(r^4 = (0.01)^4 = 0.00000001 \, \text{m}^4\).
    • Numerator: \(\pi r^4 \Delta P = \pi \times 0.00000001 \times 1000 \approx 3.14159 \times 10^{-5} \, \text{m}^4 \cdot \text{Pa}\).
    • Denominator: \(8 \eta L = 8 \times 0.001 \times 1 = 0.008 \, \text{Pa·s·m}\).
    • Flow rate in m³/s: \(Q = \frac{3.14159 \times 10^{-5}}{0.008} \approx 0.003927 \, \text{m}^3/\text{s}\).
    • Convert to output unit (L/s): \(0.003927 \times 1000 = 3.927 \, \text{L/s}\).
    • Result: \( \text{Flow Rate} = 3.9270 \, \text{L/s} \).

5. Frequently Asked Questions (FAQ)

Q: What is Poiseuille's Law?
A: Poiseuille's Law describes the volumetric flow rate of a Newtonian fluid through a cylindrical pipe under laminar flow conditions, given by \( Q = \frac{\pi r^4 \Delta P}{8 \eta L} \). It shows how flow rate depends on the pipe's radius, pressure difference, viscosity, and length.

Q: Why must radius, viscosity, and length be greater than zero?
A: These quantities must be greater than zero to represent physical properties: radius defines the pipe's size, viscosity determines the fluid's resistance to flow, and length defines the pipe's extent. Zero values would lead to division by zero or be physically meaningless.

Q: What assumptions does Poiseuille's Law make?
A: Poiseuille's Law assumes laminar flow, a Newtonian fluid (constant viscosity), a cylindrical pipe, and steady flow. It does not apply to turbulent flow, non-Newtonian fluids, or non-cylindrical geometries.

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