Natural convection

Rayleigh–Bénard cells.

Natural convection is a mechanism, or type of heat transport, in which the fluid motion is not generated by any external source (like a pump, fan, suction device, etc.) but only by density differences in the fluid occurring due to temperature gradients. In natural convection, fluid surrounding a heat source receives heat, becomes less dense and rises. The surrounding, cooler fluid then moves to replace it. This cooler fluid is then heated and the process continues, forming a convection current; this process transfers heat energy from the bottom of the convection cell to top. The driving force for natural convection is buoyancy, a result of differences in fluid density. Because of this, the presence of a proper acceleration such as arises from resistance to gravity, or an equivalent force (arising from acceleration, centrifugal force or Coriolis effect), is essential for natural convection. For example, natural convection essentially does not operate in free-fall (inertial) environments, such as that of the orbiting International Space Station, where other heat transfer mechanisms are required to prevent electronic components from overheating.

Natural convection has attracted a great deal of attention from researchers because of its presence both in nature and engineering applications. In nature, convection cells formed from air raising above sunlight-warmed land or water are a major feature of all weather systems. Convection is also seen in the rising plume of hot air from fire, oceanic currents, and sea-wind formation (where upward convection is also modified by Coriolis forces). In engineering applications, convection is commonly visualized in the formation of microstructures during the cooling of molten metals, and fluid flows around shrouded heat-dissipation fins, and solar ponds. A very common industrial application of natural convection is free air cooling without the aid of fans: this can happen on small scales (computer chips) to large scale process equipment.

Parameters

Onset

The onset of natural convection is determined by the Rayleigh number (Ra). This dimensionless number is given by

{\textbf {Ra}}={\frac {\Delta \rho gL^{3}}{D\mu }}

where

\Delta \rho

is the difference in density between the two parcels of material that are mixing

g

is the local gravitational acceleration

L

is the characteristic length-scale of convection: the depth of the boiling pot, for example

D

is the diffusivity of the characteristic that is causing the convection, and

\mu

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:

\Delta \rho =\rho _{0}\beta \Delta T

where

\rho _{0}

is the reference density, typically picked to be the average density of the medium,

\beta

is the coefficient of thermal expansion, and

\Delta T

is the temperature difference across the medium.

The general diffusivity, $D$ , is redefined as a thermal diffusivity, $\alpha$ .

D=\alpha

Inserting these substitutions produces a Rayleigh number that can be used to predict thermal convection.^[1]

{\textbf {Ra}}={\frac {\rho _{0}g\beta \Delta TL^{3}}{\alpha \mu }}

Turbulence

The tendency of a particular naturally convective system towards turbulence relies on the Grashof number (Gr).^[2]

Gr={\frac {g\beta \Delta TL^{3}}{\nu ^{2}}}

In very sticky, viscous fluids (large ν), fluid movement 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 $g\Delta \rho L^{2}/\mu$ , 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:

Gr={\frac {g\beta \Delta CL^{3}}{\nu ^{2}}}

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

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}

The value of f₄(Pr) is calculated using the following formula

f_{4}(Pr)=\left[1+\left({\frac {0.5}{Pr}}\right)^{\frac {9}{16}}\right]^{\frac {-16}{9}}

Nu is the Nusselt number and the values of Nu₀ and the characteristic length used to calculate Ra are listed below (see also Discussion):

Geometry	Characteristic length	Nu₀
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	$\pi D/2$ (D = diameter of cylinder)	0.36 $\pi$

Warning: The values indicated for the Horizontal cylinder are wrong; see discussion.

Natural Convection from a Vertical Plate

In this system heat is transferred from a vertical plate to a fluid moving parallel to it by natural convection. This will occur in any system wherein the density of the moving fluid varies with position. These phenomena will only be of significance when the moving fluid is minimally affected by forced convection.^[3]

When considering the flow of fluid is a result of heating, the following correlations can be used, assuming the fluid is an ideal diatomic, has adjacent to a vertical plate at constant temperature and the flow of the fluid is completely laminar.^[4]

Nu_m = 0.478(Gr^0.25)^[4]

Mean Nusselt Number = Nu_m = h_mL/k^[4]

Where

h_m = mean coefficient applicable between the lower edge of the plate and any point in a distance L (W/m². K)

L = height of the vertical surface (m)

k = thermal conductivity (W/m. K)

Grashof Number = Gr = $[gL^{3}(t_{s}-t_{\infty })]/v^{2}T$ ^[3]^[4]

Where

g = gravitational acceleration (m/s²)

L = distance above the lower edge (m)

t_s = temperature of the wall (K)

t∞ = fluid temperature outside the thermal boundary layer (K)

v = kinematic viscosity of the fluid (m²/s)

T = absolute temperature (K)

When the flow is turbulent different correlations involving the Rayleigh Number (a function of both the Grashof Number and the Prandtl Number) must be used.^[4]

Note that the above equation differs from the usual expression for Grashof number because the value $\beta$ has been replaced by its approximation $1/T$ , which applies for ideal gases only (a reasonable approximation for air at ambient pressure).

Pattern formation

A fluid under Rayleigh-Bénard convection: the left picture represents the thermal field and the right picture its two-dimensional Fourier transform.

Convection, especially Rayleigh-Bénard convection, where the convecting fluid is contained by two rigid horizontal plates, is a convenient example of a pattern forming system.

When heat is fed into the system from one direction (usually below), at small values it merely diffuses (conducts) from below upward, without causing fluid flow. As the heat flow is increased, above a critical value of the Rayleigh number, the system undergoes a bifurcation from the stable conducting state to the convecting state, where bulk motion of the fluid due to heat begins. If fluid parameters other than density do not depend significantly on temperature, the flow profile is symmetric, with the same volume of fluid rising as falling. This is known as Boussinesq convection.

As the temperature difference between the top and bottom of the fluid becomes higher, significant differences in fluid parameters other than density may develop in the fluid due to temperature. An example of such a parameter is viscosity, which may begin to significantly vary horizontally across layers of fluid. This breaks the symmetry of the system, and generally changes the pattern of up- and down-moving fluid from stripes to hexagons, as seen at right. Such hexagons are one example of a convection cell.

As the Rayleigh number is increased even further above the value where convection cells first appear, the system may undergo other bifurcations, and other more complex patterns, such as spirals, may begin to appear.

Mantle convection

Convection within Earth's mantle is the driving force for plate tectonics. Mantle convection is the result of a thermal gradient: the lower mantle is hotter than the upper mantle, and is therefore less dense. This sets up two primary types of instabilities. In the first type, plumes rise from the lower mantle, and corresponding unstable regions of lithosphere drip back into the mantle. In the second type, subducting oceanic plates (which largely constitute the upper thermal boundary layer of the mantle) plunge back into the mantle and move downwards towards the core-mantle boundary. Mantle convection occurs at rates of centimeters per year, and it takes on the order of hundreds of millions of years to complete a cycle of convection.

Neutrino flux measurements from the Earth's core (see kamLAND) show the source of about two-thirds of the heat in the inner core is the radioactive decay of ⁴⁰K, uranium and thorium. This has allowed plate tectonics on Earth to continue far longer than it would have if it were simply driven by heat left over from Earth's formation; or with heat produced from gravitational potential energy, as a result of physical rearrangement of denser portions of the Earth's interior toward the center of the planet (i.e., a type of prolonged falling and settling).

References

^ Donald L. Turcotte; Gerald Schubert. (2002). Geodynamics. Cambridge: Cambridge University Press. ISBN 978-0-521-66624-4.
^ Kays, William; Crawford, Michael; Weigand, Bernhard (2004). Convective Heat and Mass Transfer, 4E. McGraw-Hill Professional. ISBN 0072990732.
^ ^a ^b W. McCabe J. Smith (1956). Unit Operations of Chemical Engineering. McGraw-Hill. ISBN 0-07-044825-6.
^ ^a ^b ^c ^d ^e Bennett (1962). Momentum, Heat and Mass Transfer. McGraw-Hill. ISBN 0-07-004667-0.

[1] Donald L. Turcotte; Gerald Schubert. (2002). Geodynamics. Cambridge: Cambridge University Press. ISBN 978-0-521-66624-4.

[2] Kays, William; Crawford, Michael; Weigand, Bernhard (2004). Convective Heat and Mass Transfer, 4E. McGraw-Hill Professional. ISBN 0072990732.

[unitop-3] W. McCabe J. Smith (1956). Unit Operations of Chemical Engineering. McGraw-Hill. ISBN 0-07-044825-6.

[bennett-4] Bennett (1962). Momentum, Heat and Mass Transfer. McGraw-Hill. ISBN 0-07-004667-0.

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