Grashof number, Gr, is a nondimensional parameter used in the correlation of heat and mass transfer due to thermally induced natural convection at a solid surface immersed in a fluid. It is defined as
g = acceleration due to gravity, m s−2
l = representative dimension, m
ξ = coefficient of expansion of the fluid, K−1
ΔT = temperature difference between the surface and the bulk of the fluid, K
ν = kinematic viscosity of the fluid, m2s−1 .
The significance of the Grashof number is that it represents the ratio between the buoyancy force due to spatial variation in fluid density (caused by temperature differences) to the restraining force due to the viscosisty of the fluid.
The form of the Grashof number can be derived by considering the forces on a small element of fluid of volume l3.
The buoyancy force, Fb, on this element has the magnitude gl3Δρ, where Δρ is the difference in density between the element and the surrounding fluid. The order of magnitude of the viscous force, Fv, on the element is ηul, where η is the fluid viscosity, and u the velocity of the element relative to the surrounding fluid. Hence,
The order of magnitude of the velocity u may be obtained by equating viscous and momentum forces, i.e.,
Substituting this value into the ratio of buoyancy to viscous forces
and using the relationship
Since Reynolds number, Re, represents the ratio of momentum to viscous forces (see Reynolds Number) the relative magnitudes of Gr and Re are an indication of the relative importance of natural and forced convection in determining heat transfer. Forced convection effects are usually insignificant when Gr/Re2 >> 1 and conversely natural convection effects may be neglected when Gr/Re2 << 1. When the ratio is of the order of one, combined effects of natural and forced convection have to be taken into account.
Hewitt G. F, Shires, G. L. and Bott T. R. (1994) Process Heat Transfer, CRC Press, Boca Raton, FL, USA.