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Hagen Number

 

Hagen number formula
\( Hg \;=\;   \dfrac{ 1 }{ \rho }  \cdot  \dfrac{ d p }{ d x}  \cdot  \dfrac{ l^3 }{ \nu^2}  \) 
Symbol English Metric
\( Hg \) = Hagen Number \( dimensionless \) \( dimensionless \)
\( \rho \)  (Greek symbol rho) = Fluid Density \(lbm\;/\;ft^3\) \(kg\;/\;m^3\)
\( d p \) = Pressure Differential \(lbf\;/\;in^2\) \(Pa\)
\( \frac {d p}{d x}  \) = Pressure Gradient \(psi\;/\;ft\) \(Pa\;/\;m\)
\(  d x  \) = Distance Between Pressure Centers \(t\) \(m\)
\( l \) = Length \(ft\) \(m\)
\( \nu \)  (Greek symbol nu) = Kinematic Viscosity \(ft^2\;/\;sec\) \(m^2\;/\;s\)

Hagen number. abbrviated as Hg, a dimensionless number, often used to characterize flow driven by a pressure gradient, particularly in laminar flow through pipes or channels.  It’s closely tied to the Hagen-Poiseuille flow, which describes the steady, laminar flow of a viscous fluid through a cylindrical tube.  Unlike some other dimensionless numbers, it’s not as universally standardized in its definition or use, but it typically emerges in contexts involving pressure-driven flows.

Hagen Number Interpretation
Low Hagen Number (Hg << 1)  -  Viscous forces dominate over the pressure gradient.  The fluid moves slowly because the pressure difference isn’t enough to overcome the internal friction effectively.  This might happen in very viscous fluids, narrow pipes, or weak pressure gradients.
Hagen Number (Hg ≈ 1)  -  The pressure gradient and viscous forces are in a balanced interplay, typical of many practical laminar flow scenarios (like in microfluidics or slow flow through capillaries).
High Hagen Number (Hg >> 1)  -  The pressure gradient is strong relative to viscous resistance.  This suggests a significant driving force pushing the fluid through the pipe, leading to higher flow rates for a given viscosity and pipe size.  The flow is still laminar (assuming Hg doesn’t push the system into turbulence), but it’s more "pressure-dominated."
 
Relation to other Dimensionless Numbers
Bejan Number  -  In some cases, the Hagen Number aligns with the Bejan Number, which is a dimensionless pressure drop.  They coincide when the characteristic length equals the flow length, though their physical meanings differ slightly (pressure gradient vs. pressure drop).
Reynolds Number  -  While the Reynolds Number compares inertial to viscous forces, the Hagen Number focuses on the pressure gradient’s role, offering a complementary perspective in forced flow scenarios.

The Hagen number is crucial in understanding fluid flow in microfluidics, capillary tubes, and porous media, where the interaction between viscous and surface tension forces becomes important.  It's also relevant in applications involving liquid transport, such as in certain medical devices, inkjet printing, and oil reservoir modeling.

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