Editing Convection (section)

==Mathematical models of convection==
A number of dimensionless terms have been derived to describe and predict convection, including the [[Archimedes number]], [[Grashof number]], [[Richardson number]], and the [[Rayleigh number]].

In cases of mixed convection (natural and forced occurring together) one would often like to know how much of the convection is due to external constraints, such as the fluid velocity in the pump, and how much is due to natural convection occurring in the system.

The relative magnitudes of the [[Grashof number]] and the square of the [[Reynolds number]] determine which form of convection dominates. If <math>\rm Gr/Re^2 \gg 1 </math>, [[forced convection]] may be neglected, whereas if <math>\rm Gr/Re^2 \ll 1 </math>, natural convection may be neglected. If the ratio, known as the [[Richardson number#Thermal convection|Richardson number]], is approximately one, then both forced and natural convection need to be taken into account.

===Onset===
{{See also|Heat transfer}}
The onset of natural convection is determined by the [[Rayleigh number]] ('''Ra'''). This [[dimensionless number]] is given by

:<math>\textbf{Ra} = \frac{\Delta\rho g L^3}{D\mu}</math>

where

*<math>\Delta \rho</math> is the difference in density between the two parcels of material that are mixing
*<math>g</math> is the local [[gravitational acceleration]]
*<math>L</math> is the characteristic length-scale of convection: the depth of the boiling pot, for example
*<math>D</math> is the [[diffusivity]] of the characteristic that is causing the convection, and
*<math>\mu</math> is the [[dynamic viscosity]].

Natural convection will be more likely and/or more rapid with a greater variation in density between the two fluids, a larger acceleration due to gravity that drives the convection, and/or a larger distance through the convecting medium. Convection will be less likely and/or less rapid with more rapid diffusion (thereby diffusing away the gradient that is causing the convection) and/or a more viscous (sticky) fluid.

For thermal convection due to heating from below, as described in the boiling pot above, the equation is modified for thermal expansion and thermal diffusivity. Density variations due to thermal expansion are given by:

:<math>\Delta\rho=\rho_0 \beta \Delta T</math>

where

*<math>\rho_0</math> is the reference density, typically picked to be the average density of the medium,
*<math>\beta</math> is the [[coefficient of thermal expansion]], and
*<math>\Delta T</math> is the temperature difference across the medium.

The general diffusivity, <math>D</math>, is redefined as a [[thermal diffusivity]], <math>\alpha</math>.

:<math>D=\alpha</math>

Inserting these substitutions produces a Rayleigh number that can be used to predict thermal convection.<ref>{{cite book|isbn=978-0-521-66624-4|author1=Donald L. Turcotte |author2=Gerald Schubert. |year=2002|publisher=Cambridge University Press|location=Cambridge|title=Geodynamics}}</ref>

:<math>\textbf{Ra} = \frac{\rho_0 g \beta \Delta T L^3}{\alpha \mu}</math>

===Turbulence===
The tendency of a particular naturally convective system towards turbulence relies on the [[Grashof number]] (Gr).<ref>{{cite book |author1=Kays, William |author2=Crawford, Michael |author3=Weigand, Bernhard | title=Convective Heat and Mass Transfer, 4E | publisher=McGraw-Hill Professional | year=2004 | isbn=978-0072990737}}</ref>

:<math> Gr= \frac{g \beta \Delta T L^3}{\nu^2} </math>

In very sticky, viscous fluids (large ''&nu;''), fluid motion is restricted, and natural convection will be non-turbulent.

Following the treatment of the previous subsection, the typical fluid velocity is of the order of <math>g \Delta \rho L^2 / \mu</math>, up to a numerical factor depending on the geometry of the system. Therefore, Grashof number can be thought of as [[Reynolds number]] with the velocity of natural convection replacing the velocity in Reynolds number's formula. However In practice, when referring to the Reynolds number, it is understood that one is considering forced convection, and the velocity is taken as the velocity dictated by external constraints (see below).

===Behavior===
The [[Grashof number]] can be formulated for natural convection occurring due to a [[concentration gradient]], sometimes termed thermo-solutal convection.  In this case, a concentration of hot fluid diffuses into a cold fluid, in much the same way that ink poured into a container of water diffuses to dye the entire space. Then:

:<math> Gr= \frac{g \beta \Delta C L^3}{\nu^2} </math>

Natural convection is highly dependent on the geometry of the hot surface, various correlations exist in order to determine the heat transfer coefficient. 
A general correlation that applies for a variety of geometries is

: <math>Nu = \left[Nu_0^\frac{1}{2} + Ra^ \frac{1}{6} \left(\frac {f_4\left(Pr\right)}{300}\right)^\frac{1}{6} \right]^2 </math>

The value of f<sub>4</sub>(Pr) is calculated using the following formula

: <math>f_4(Pr)= \left[1+ \left ( \frac {0.5}{Pr} \right )^\frac{9}{16} \right]^\frac{-16}{9}</math>

Nu is the [[Nusselt number]] and the values of Nu<sub>0</sub> and the characteristic length used to calculate Re are listed below (see also Discussion):

{| class="wikitable"
|-
! '''Geometry'''
! '''Characteristic length'''
! '''Nu<sub>0</sub>'''
|-
| Inclined plane
| x (Distance along plane)
| 0.68
|-
| Inclined disk
| 9D/11 (D = diameter)
| 0.56
|-
| Vertical cylinder
| x (height of cylinder)
| 0.68
|-
| Cone
| 4x/5 (x = distance along sloping surface)
| 0.54
|-
| Horizontal cylinder
| <math>\pi D/2</math> (D = diameter of cylinder)
| 0.36<math>\pi</math>
|}
'''Warning''': The values indicated for the '''Horizontal cylinder''' are '''wrong'''; see discussion.