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Concept
Grashof number
Summary
In fluid mechanics (especially fluid thermodynamics), the Grashof number (Gr, after Franz Grashof) is a dimensionless number which approximates the ratio of the buoyancy to viscous forces acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number (Re).
Free convection is caused by a change in density of a fluid due to a temperature change or gradient. Usually the density decreases due to an increase in temperature and causes the fluid to rise. This motion is caused by the buoyancy force. The major force that resists the motion is the viscous force. The Grashof number is a way to quantify the opposing forces.
The Grashof number is:
for vertical flat plates
for pipes
for bluff bodies
where:
g is gravitational acceleration due to Earth
β is the coefficient of volume expansion (equal to approximately 1/T for ideal gases)
T_s is the surface temperature
T_∞ is the bulk temperature
L is the vertical length
D is the diameter
ν is the kinematic viscosity.
The L and D subscripts indicate the length scale basis for the Grashof number.
The transition to turbulent flow occurs in the range 10^8 < 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, that is, in the range 10^3 < Gr_L < 10^6.
There is an analogous form of the Grashof number used in cases of natural convection mass transfer problems. In the case of mass transfer, natural convection is caused by concentration gradients rather than temperature gradients.
where
and:
g is gravitational acceleration due to Earth
C_a,s is the concentration of species a at surface
C_a,a is the concentration of species a in ambient medium
L is the characteristic length
ν is the kinematic viscosity
ρ is the fluid density
C_a is the concentration of species a
T is the temperature (constant)
p is the pressure (constant).
The Rayleigh number, shown below, is a dimensionless number that characterizes convection problems in heat transfer.
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