Viscous Flow Rate Calculator Formula

Understand the math behind the viscous flow rate calculator. Each variable explained with a worked example.

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Formulas Used

Flow Rate

flow_rate = pi * pow(pipe_radius, 4) * pressure_drop / (8 * viscosity * pipe_length)

Flow Rate (L/min)

flow_rate_lpm = pi * pow(pipe_radius, 4) * pressure_drop / (8 * viscosity * pipe_length) * 60000

Variables

VariableDescriptionDefault
pipe_radiusPipe Radius(m)0.01
pressure_dropPressure Drop(Pa)1000
viscosityDynamic Viscosity(Pa s)0.001
pipe_lengthPipe Length(m)1

How It Works

Hagen-Poiseuille Flow

For steady, laminar, incompressible flow through a long circular pipe, the flow rate depends strongly on the pipe radius.

Formula

Q = pi R^4 dP / (8 mu L)

  • *R* = pipe radius
  • *dP* = pressure drop along the pipe
  • *mu* = dynamic viscosity
  • *L* = pipe length

    • The R^4 dependence means doubling the radius increases flow by 16 times.

    Worked Example

    Water (mu = 0.001 Pa s) through a 1 cm radius, 1 m long pipe with 1 kPa pressure drop.

    pipe_radius = 0.01pressure_drop = 1000viscosity = 0.001pipe_length = 1
    1. 01Q = pi * (0.01)^4 * 1000 / (8 * 0.001 * 1)
    2. 02R^4 = 1e-8
    3. 03Q = pi * 1e-8 * 1000 / 0.008
    4. 04Q = pi * 1e-5 / 0.008 = 3.927e-3 m3/s
    5. 05Q = 235.6 L/min

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