# Grashof number

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The Grashof number 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.

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

The transition to turbulent flow occurs in the range 108 < GrL < 109 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.

where

and

## Contents

### Derivation of Grashof Number

The first step to deriving the Grashof Number Gr is manipulating the volume expansion coefficient, β as follows:

$\mathrm{\beta}=\frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_p

\mathrm=\frac{-1}{\rho}\left(\frac{\partial\rho}{\partial T}\right)_p$

This partial relation of the volume expansion coefficient, β with respect to fluid density, ρ and constant pressure can be rewritten as

ρ = ρo(1 − βΔT)

and

ρo - bulk fluid density ρ - boundary layer density ΔT = (TTo) - temperature difference between boundary layer and bulk fluid

There are two different ways to find the Grashof Number from this point. One involves the energy equation while the other incorporates the buoyant force due to the difference in density between the boundary layer and bulk fluid.

### Energy Equation

This discussion involving the energy equation is with respect to rotationally symmetric flow. This analysis will take into consideration the effect of gravitational acceleration on flow and heat transfer. The mathematical equations to follow apply both to rotational symmetric flow as well as two-dimensional planar flow.

$\mathrm{\frac{\partial}{\partial s}}(\rho u r_o^{n})+{\frac{\partial}{\partial y}}(\rho \nu r_o^{n})=0$

s - rotational direction u - tangential velocity y - planar direction ν - normal velocity ro - radius

This equation expands to the following with the addition of physical fluid properties:

$\mathrm{\rho}(u \frac{\partial u}{\partial s}+\nu \frac{\partial u}{\partial y})=\frac{\partial}{\partial y}(\mu \frac{\partial u}{\partial y})-\frac{d \mathrm{p}}{d s}+\rho g$