Hagen number

Physical key figure
Surname	Hagen number
Formula symbol	${\displaystyle {\mathit {Hg}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Hg}}=-{\frac {1}{\rho }}{\frac {\mathrm {d} p}{\mathrm {d} z}}{\frac {L^{3}}{\nu ^{2}}}}$
${\displaystyle \rho }$ density ${\displaystyle {\frac {\mathrm {d} p}{\mathrm {d} z}}}$ Pressure gradient ${\displaystyle L}$ characteristic length ${\displaystyle \nu }$ kinematic viscosity
Named after	Gotthilf Hagen
scope of application	forced flow

The Hagen number (named after Gotthilf Hagen ) is a dimensionless number for forced flow . It can be viewed as a dimensionless pressure gradient : ${\displaystyle {\mathit {Hg}}}$

${\displaystyle {\mathit {Hg}}=-{\frac {1}{\rho }}\cdot {\frac {\mathrm {d} p}{\mathrm {d} z}}\cdot {\frac {L^{3}}{\nu ^{2}}}}$

With

• ${\displaystyle \rho }$ Density of the fluid
• ${\displaystyle {\frac {\mathrm {d} p}{\mathrm {d} z}}}$Pressure gradient along the z- axis that drives the forced flow
• ${\displaystyle L}$ characteristic length
• ${\displaystyle \nu }$ kinematic viscosity .

The following applies to free currents

${\displaystyle {\frac {\mathrm {d} p}{\mathrm {d} z}}=g\cdot \rho \cdot \beta \cdot (T_{\mathrm {s} }-T_{\infty })}$

With

• ${\displaystyle g}$ Gravitational acceleration ( ) ${\displaystyle \approx 9{,}81\;\mathrm {\tfrac {m}{s^{2}}} }$
• ${\displaystyle \beta }$ Coefficient of thermal expansion
• ${\displaystyle T_{\mathrm {s} }}$ temperature
• ${\displaystyle T_{\infty }}$ Resting temperature

so that in this case the Hagen number merges into the Grashof number .