View source for Grashof number

The '''Grashof number''' ('''Gr''') is a [[dimensionless number]] that arises in natural (or free) [[convection]]. Free convection is caused by a change in the density of a fluid due to a temperature change or gradient, which drives fluid flow due to a buoyancy force. The flow is resisted by viscous forces. The definition of the Grashof number is the ratio of this buoyancy to viscous force, and is analogous to the Reynolds number.

==Definition==
The Grashof number is:

For Vertical Flat Plates:
{{NumbEq|1|
<math> \mathrm{Gr}_L = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2}\ </math>}}

For Pipes:
{{NumbEq|2|
<math> \mathrm{Gr}_D = \frac{g \beta (T_s - T_\infty ) D^3}{\nu ^2}\ </math>}}

For Bluff Bodies:
{{NumbEq|3|
<math> \mathrm{Gr}_D = \frac{g \beta (T_s - T_\infty ) D^3}{\nu ^2}\ </math>}}

where
*{{Math|g}} is the gravitational constant
*{{Math|\beta}} is the coefficient of thermal expansion (equal to <math> 1/T </math> for ideal gases)
*{{Math|T_s}} is the surface temperature
*<math> T_{\infty} </math> is the bulk temperature
*{{Math|L}} is the vertical length
*{{Math|D}} is the diameter
*{{Math|\nu}} is the kinematic viscosity {{Math|\mu / \rho}}.

The {{Math|L}} and {{Math|D}} subscripts indicate the length scale basis for the Grashof number.

==Relationship to the Rayleigh and Reynolds number==
The [[Rayleigh number]], shown below, is a dimensionless number that characterizes convection problems in heat transfer. A critical value exists for the [[Rayleigh number]], above which fluid motion occurs.

{{NumbEq|4|
<math>\mathrm{Ra} = \mathrm{Gr}\mathrm{Pr}</math>}}

The ratio of the Grashof number to the square of the [[Reynolds number]] may be used to determine if forced or free convection may be neglected for a system, or if there's a combination of the two. If the ratio is much less than one, then free convection may be ignored. If the ratio is much greater than one, forced convection may be ignored. Otherwise, the regime is combined forced and free convection.

Forced convection may be ignored:
{{NumbEq|5|
<math>\frac{\mathrm{Gr}}{\mathrm{Re}^2} \gg 1 </math>}}

Combined forced and free convection:
{{NumbEq|6|
<math>\frac{\mathrm{Gr}}{\mathrm{Re}^2} \approx 1 </math>}}

Free convection may be neglected:
{{NumbEq|7|
<math>\frac{\mathrm{Gr}}{\mathrm{Re}^2} \ll 1 </math>}}

[[Category:Variable]]
[[Category:Dimensionless number]]


In the process of heat transfer, the determination of heat transfer is dependent on the relative magnitude of Grashof number and Reynolds number for the forced convection heat transfer process and free convection heat transfer process as ratio of momentum force to viscous force is Reynolds number and ratio of buoyancy force to viscous force is Grashof number <ref name ="one" [Li, X. F. (2013). Grashof Number Effects on Nanofluids in Natural Convection Heat Transfer. Applied Mechanics and Materials , 43 - 48. </ref>.