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Grashof Number

 

Grashof Number for vertical flat places formula
\( Gr \;=\;  \dfrac{ g \cdot l^3 \cdot \alpha_c \cdot ( T_s - T_{\infty}) }{ \nu^2 }\)
Symbol English Metric
\(\large{ Gr }\) = Grashof Number \(dimensionless\) \(dimensionless\)
\(\large{ T_{\infty} }\) = Bulk Temperature \(F\) \(C\)
\(\large{ g }\) = Fluid Gravitational Acceleration \(ft \;/\; sec^2\) \(m \;/\; s^2\)
\(\large{ \nu }\)  (Greek symbol nu) = Fluid Kinematic Viscosity \(ft^2 \;/\; sec\) \(m^2 \;/\; s\)
\(\large{ l }\) = Pipe Vertical Length \(ft\) \(m\)
\(\large{ T_s }\) = Surface Temperature \(F\) \(C\)
\(\large{ \alpha_c }\)  (Greek symbol alpha) = Pipe Thermal Expansion Coefficient \(in \;/\; in\;F\) \(mm \;/\; mm\;C\)

Grashof number, abbreviated as Gr, a dimensionless number, is used in fluid dynamics and heat transfer to characterize the relative importance of buoyancy forces to viscous forces in a fluid flow or heat transfer process.  It is particularly relevant in natural convection, where fluid motion is driven by temperature differences, and buoyancy plays a significant role.  The Grashof number helps determine the dominant mode of heat transfer within a fluid syste

Grashof Number for bulk bodies and pipes formula
\( Gr \;=\; \dfrac{   g \cdot l^3 \cdot \alpha_c \cdot ( T_s^{\nu^2} - T_{\infty} ) }{  \nu^2 } \)
Symbol English Metric
\(\large{ Gr }\) = Grashof Number \(dimensionless\) \(dimensionless\)
\(\large{ T_{\infty} }\) = Bulk Temperature \(F\) \(C\)
\(\large{ g }\) = Fluid Gravitational Acceleration \(ft \;/\; sec^2\) \(m \;/\; s^2\)
\(\large{ \nu }\)  (Greek symbol nu) = Fluid Kinematic Viscosity \(ft^2 \;/\; sec\) \(m^2 \;/\; s\)
\(\large{ l }\) = Pipe Vertical Length \(ft\) \(m\)
\(\large{ T_s }\) = Fluid Surface Temperature \(F\) \(C\)
\(\large{ \alpha_c }\)  (Greek symbol alpha) = Pipe Thermal Expansion Coefficient \(in \;/\; in\;F\) \(mm \;/\; mm\;C\)
Grashof Number Interpretation
Low Grashof Number (Gr << 1)  -  Viscous forces dominate over buoyancy forces.  The fluid stays mostly still, and any motion due to temperature differences is weak or negligible.  Heat transfer happens mainly by conduction rather than convection.  This occurs with small length scales, low \(\Delta T\), or high viscosity (a thin layer of thick oil heated slightly).
Intermediate Grashof Number (Gr ≈ 1)  -  There’s a crossover range where convection starts to kick in.  For vertical plates, \(Gr \approx 10^4\) often marks the shift from laminar to turbulent convection, though this depends on the system geometry and the Rayleigh Number (\(Ra = Gr \cdot Pr\) , which includes thermal diffusivity effects).
High Grashof Number (Gr >> 1)  -  Buoyancy forces dominate over viscous forces.  The fluid moves significantly due to density differences, hot fluid rises, cold fluid sinks, creating strong natural convection currents.  This happens with large length scales, big \(\Delta T\), or low viscosity (air around a hot stove).  Heat transfer is enhanced by this motion.

The Grashof Number often pairs with the Prandtl Number (Pr) to form the Rayleigh Number (Ra), which fully describes natural convection by including thermal diffusion.  Gr alone focuses on the buoyancy-viscosity balance.

The Grashof number is commonly used in the analysis and design of heat exchangers, cooling systems, and other situations where fluid motion is driven by temperature differences.  It helps engineers and scientists predict when natural convection will be a significant factor in a given system and how it will affect heat transfer or fluid flow.

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