Hagen–Poiseuille Law Formula |
||
|
\( Q \;=\; \dfrac{ \pi \cdot r^4 \cdot \Delta p }{ 8 \cdot \mu \cdot L }\) (Hagen–Poiseuille Law) \( r \;=\; \left( \dfrac{ \pi \cdot \Delta p }{ 8 \cdot \mu \cdot L \cdot Q} \right)^{ \frac{1}{4} } \) \( \Delta p \;=\; \dfrac{ 8 \cdot \mu \cdot L \cdot Q }{ \pi \cdot r^4 }\) \( \mu \;=\; \dfrac{ \pi \cdot r^4 \cdot \Delta p }{ 8 \cdot L \cdot Q }\) \( L \;=\; \dfrac{ \pi \cdot r^4 \cdot \Delta p }{ 8 \cdot \mu \cdot Q }\) |
||
| 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 is a principle in fluid dynamics that governs the flow of viscous, incompressible fluids through cylindrical pipes under laminar flow conditions. It provides a mathematical relationship between the flow rate of the fluid and several key parameters, including the fluid's viscosity, the pipe's dimensions, and the pressure difference driving the flow. This law is particularly valuable for understanding fluid behavior in scenarios where the flow is smooth and orderly, without turbulent eddies or chaotic motion.
A central aspect of the Hagen–Poiseuille law is its emphasis on the relationship between the flow rate and the pipe's radius. The flow rate is directly proportional to the fourth power of the radius, meaning that even small changes in the pipe's diameter result in substantial variations in the fluid's flow. This sensitivity to radius is crucial in various applications, from designing medical devices to analyzing blood flow within the circulatory system. Furthermore, the law states that the flow rate is directly proportional to the pressure difference between the pipe's ends and inversely proportional to the fluid's viscosity and the pipe's length. This means that a greater pressure difference drives a faster flow, while higher viscosity or a longer pipe impedes it.
Hagen–Poiseuille Law Formula |
||
|
\( \eta \;=\; \dfrac{ ( p_o - p_i ) \cdot \pi \cdot r^4 \cdot \rho }{ 8 \cdot \dot m_f \cdot L }\) (Hagen–Poiseuille Law) \( p_o \;=\; \left( \dfrac{ 8 \cdot \dot m_f \cdot L \cdot \eta }{ \pi \cdot r^4 \cdot \rho } \right) + p_i \) \( p_i \;=\; p_o - \left( \dfrac{ 8 \cdot \dot m_f \cdot L \cdot \eta }{ \pi \cdot r^4 \cdot \rho } \right) \) \( r \;=\; \left( \dfrac{ 8 \cdot \dot m_f \cdot L \cdot \eta }{ ( p_o - p_i ) \cdot \pi \cdot \rho } \right)^{ \frac{1}{4} } \) \( \rho \;=\; \dfrac{ 8 \cdot \dot m_f \cdot L \cdot \eta }{ ( p_o - p_i ) \cdot \pi \cdot r^4 } \) \( \dot m_f \;=\; \dfrac{ ( p_o - p_i ) \cdot \pi \cdot r^4 \cdot \rho }{ 8 \cdot \eta \cdot L }\) \( L \;=\; \dfrac{ ( p_o - p_i ) \cdot \pi \cdot r^4 \cdot \rho }{ 8 \cdot \dot m_f \cdot \eta }\) |
||
| 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 | \(ft\) | \(m\) |