Grashof number

The Grashof number $mathrm{Gr}$ is a dimensionless number in fluid dynamics and Heat Transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in the study of situations involving natural convection. It is named after the German engineer Franz Grashof.

: $mathrm{Gr}_L = frac{g eta (T_s - T_infty ) L^3}{
u ^2},$ for vertical flat plates

: $mathrm{Gr}_D = frac{g eta (T_s - T_infty ) D^3}{
u ^2},$ for pipes: $mathrm{Gr}_D = frac{g eta (T_s - T_infty ) D^3}{
u ^2},$ for bluff bodies

where the L and D subscripts indicates the length scale basis for the Grashof Number.

: "g" = acceleration due to Earth's gravity: "β" = volumetric thermal expansion coefficient (equal to approximately 1/T, for ideal fluids, where T is absolute temperature): "T"_"s" = source temperature: "T"_"∞" = film temperature: "L" = length: "D" = diameter: "ν" = kinematic viscosity

The transition to turbulent flow occurs in the range $10^8 < mathrm{Gr}_L < 10^9$ for natural convection from vertical flat plates. At higher Grashof numbers, the boundary layer is turbulent; at lower Grashof numbers, the boundary layer is laminar.

The product of the Grashof number and the Prandtl number gives the Rayleigh number, a dimensionless number that characterizes convection problems in heat transfer.

There is an analogous form of the Grashof number used in cases of natural convection mass transfer problems.

: $mathrm{Gr}_c = frac{g eta^* (C_{a,s} - C_{a,a} ) L^3}{
u^2}$

where

: $eta^* = -frac{1}{
ho} left ( frac{partial
ho}{partial C_a}
ight )_{T,p}$

and

: "g" = acceleration due to Earth's gravity: "C"_"a,s" = concentration of species "a" at surface: "C"_"a,a" = concentration of species "a" in ambient medium : "L" = characteristic length: "ν" = kinematic viscosity: "ρ" = fluid density: "C"_"a" = concentration of species "a": "T" = constant temperature: "p" = constant pressure

ee also

References

* Jaluria, Yogesh. "Natural Convection Heat and Mass Transfer" (New York: Pergamon Press, 1980).

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