Hagen–Poiseuille law, also called Hagen–Poiseuille equation, Poiseuille equation, or Poiseuille law, describes the flow of a viscous, incompressible fluid through a cylindrical tube, such as petroleum through a pipe. It relates the pressure drop across the tube to the flow rate, fluid viscosity, and tube dimensions. In simple terms, it’s a way to calculate how much fluid (like petroleum) moves through a pipe under specific conditions.
Hagen–Poiseuille Law Formula |
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\( Q \;=\; \dfrac{ \pi \cdot r^4 \cdot \Delta p }{ 8 \cdot \mu \cdot L }\) (Hagen–Poiseuille Law) \( r \;=\; \left( \dfrac{ \pi \cdot \Delta p }{ 8 \cdot \mu \cdot L \cdot Q} \right)^{ \frac{1}{4} } \) \( \Delta p \;=\; \dfrac{ 8 \cdot \mu \cdot L \cdot Q }{ \pi \cdot r^4 }\) \( \mu \;=\; \dfrac{ \pi \cdot r^4 \cdot \Delta p }{ 8 \cdot L \cdot Q }\) \( L \;=\; \dfrac{ \pi \cdot r^4 \cdot \Delta p }{ 8 \cdot \mu \cdot Q }\) |
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| Symbol | English | Metric |
| \( Q \) = Volumetric Flow Rate | \(ft^3\;/\;sec\) | \(m^3\;/\;s\) |
| \( \pi \) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( r \) = Pipe Inside Radius | \(in\) | \(mm\) |
| \( \Delta p \) = Pressure Loss (psi) between the Ends of the Pipe | \(lbf\;/\;in^2\) | \(Pa\) |
| \( \mu \) (Greek symbol mu) = Dynamic Viscosity (for Petroleum, this Depends on its Type and Temperature) | \(lbf-sec\;/\;ft^2\) | \(Pa-s\) |
| \( L \) = Pipe Length | \(ft\) | \(m\) |
Hagen–Poiseuille Law Formula |
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\( \eta \;=\; \dfrac{ ( p_o - p_i ) \cdot \pi \cdot r^4 \cdot \rho }{ 8 \cdot \dot m_f \cdot L }\) (Hagen–Poiseuille Law) \( p_o \;=\; \left( \dfrac{ 8 \cdot \dot m_f \cdot L \cdot \eta }{ \pi \cdot r^4 \cdot \rho } \right) + p_i \) \( p_i \;=\; p_o - \left( \dfrac{ 8 \cdot \dot m_f \cdot L \cdot \eta }{ \pi \cdot r^4 \cdot \rho } \right) \) \( r \;=\; \left( \dfrac{ 8 \cdot \dot m_f \cdot L \cdot \eta }{ ( p_o - p_i ) \cdot \pi \cdot \rho } \right)^{ \frac{1}{4} } \) \( \rho \;=\; \dfrac{ 8 \cdot \dot m_f \cdot L \cdot \eta }{ ( p_o - p_i ) \cdot \pi \cdot r^4 } \) \( \dot m_f \;=\; \dfrac{ ( p_o - p_i ) \cdot \pi \cdot r^4 \cdot \rho }{ 8 \cdot \eta \cdot L }\) \( L \;=\; \dfrac{ ( p_o - p_i ) \cdot \pi \cdot r^4 \cdot \rho }{ 8 \cdot \dot m_f \cdot \eta }\) |
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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 | \(ft\) | \(m\) |
