Hagen–Poiseuille Law
Hagen–Poiseuille law, also called Hagen–Poiseuille equation, Poiseuille equation, or Poiseuille law, gives the pressure loss in a fluid flowing through a long cylindrical pipe. It describes the laminar flow of a Newtonian fluid through a long, straight, cylindrical pipe of constant cross-section. It relates the pressure drop across the pipe to the flow rate and the properties of the fluid and pipe.
The law assumes laminar flow, which is characterized by smooth, ordered flow patterns and is observed at low Reynolds numbers. The equation also assumes that the fluid is incompressible, the flow is steady-state, and the pipe is long and straight with a constant cross-sectional area. The Hagen-Poiseuille law is commonly used in engineering and fluid dynamics to calculate pressure drop and flow rate in pipes, as well as to determine the properties of fluids from pressure flow rate data.
Hagen–Poiseuille Law formula |
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\( \Delta p \;=\; 8 \; \mu \; L \; Q \;/\; \pi \; r^4 \) (Hagen–Poiseuille Law) \( \mu \;=\; \pi \; r^4 \; \Delta p \;/\; 8 \; L \; Q \) \( L \;=\; \pi \; r^4 \; \Delta p \;/\; 8 \; \mu \; Q \) \( Q \;=\; \pi \; r^4 \; \Delta p \;/\; 8 \; \mu \; L \) \( r \;=\; (\; \pi \; \Delta p \;/\; 8 \; \mu \; L \; Q \;)^{ \frac{1}{4} } \) |
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| Symbol | English | Metric |
| \( \Delta p \) = Pressure Loss (psi) | \(lbf\;/\;in^2\) | \(Pa\) |
| \( \mu \) (Greek symbol mu) = Dynamic Viscosity | \(lbf-sec\;/\;ft^2\) | \(Pa-s\) |
| \( L \) = Pipe Length | \(ft\) | \(m\) |
| \( Q \) = Volumetric Flow Rate | \(ft^3\;/\;sec\) | \(m^3\;/\;s\) |
| \( \pi \) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( r \) = Pipe Inside Radius | \(in\) | \(mm\) |
Hagen–Poiseuille Law formula |
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\( \eta \;=\; ( p_o - p_i ) \; \pi \; r^4 \;\rho \;/\; 8 \; \dot m_f \; L \) (Hagen–Poiseuille Law) \( p_o \;=\; ( \;8 \; \dot m_f \; L \; \eta \;/\; \pi \; r^4 \; \rho \; ) + p_i \) \( p_i \;=\; p_o - ( \;8 \; \dot m_f \; L \; \eta \;/\; \pi \; r^4 \; \rho \;) \) \( r \;=\; ( \;8 \; \dot m_f \; L \; \eta \;/\; ( p_o - p_i ) \; \pi \; \rho\; )^{ \frac{1}{4} } \) \( \rho \;=\; 8 \; \dot m_f \; L \; \eta \;/\; ( p_o - p_i ) \; \pi \; r^4 \) \( \dot m_f \;=\; ( p_o - p_i ) \; \pi \; r^4 \;\rho \;/\; 8 \; \eta \; L \) \( L \;=\; ( p_o - p_i ) \; \pi \; r^4 \;\rho \;/\; 8 \; \dot m_f \; \eta \) |
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| Symbol | English | Metric |
| \( \eta \) (Greek symbol eta) = Medium Viscosity | \(lbf - sec \;/\; ft^2\) | \( Pa - s \) |
| \( p_o \) = Output Pressure (psi) | \(lbf \;/\; in^2\) | \(Pa\) |
| \( p_i \) = Input Pressure (psi) | \(lbf \;/\; in^2\) | \(Pa\) |
| \( \pi \) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( r \) = Pipe Inside Radius | \(in\) | \(mm\) |
| \( \rho \) (Greek symbol rho) = Fluid Density | \(lbm \;/\; ft^3\) | \(kg \;/\; m^3\) |
| \( \dot m_f \) = Mass Flow Rate | \(lbm \;/\; sec\) | \(kg \;/\; s\) |
| \( L \) = Pipe Length | \(ft\) | \(m\) |
Tags: Pipe Pressure Laws of Physics Laws of Fluid Dynamics Laminar Flow Reservoir