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conductive heat

Convective heat transfer

Convective heat transfer is a mechanism of heat transfer occurring because of bulk motion (observable movement) of fluids. This can be contrasted with conductive heat transfer, which is the transfer of energy by vibrations at a molecular level through a solid or fluid, and radiative heat transfer, the transfer of energy through electromagnetic waves.

As convection is dependent on the bulk movement of a fluid it can only occur in liquids, gases and multiphase mixtures.

Convective heat transfer is split into two categories: natural (or free) convection and forced (or advective) convection, also known as heat advection.

Natural convective heat transfer

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 gravity or an equivalent force (arising from the equivalence principle, such as acceleration, centrifugal force or Coriolis force) is essential for natural convection. For example, natural convection does not operate as it does on Earth in the freefall environment 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 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 Coriolus 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.

Mathematically, the tendency of a particular system towards natural convection relies on the Grashof number (Gr), which is a ratio of buoyancy force and viscous force.

Gr= frac{g beta Delta T L^3}{nu^2}

The parameter beta is the volume expansivity (K-1), g is acceleration due to gravity, DeltaT is the temperature difference between the hot surface and the bulk fluid (K), L is the characteristic length (this depends on the object) and ν is the viscosity.

For liquids, values of beta are tabulated. Additionally beta can be calculated from:

beta =frac{1}{V} frac {d V}{dT}= frac{1}{v} frac {d v}{dT}= -frac{1}{rho} frac {d rho}{dT} (K-1)

For an ideal gas, this number may be simply found:

PV = nRT
PV = frac{m}{mol. weight}RT
frac{m}{V} = rho = frac{P times mol. weight}{RT}
frac{drho}{dT} = - frac{P times mol. weight}{R} frac{1}{T^2}
beta = -frac{1}{rho} frac{drho}{dT} = -left (frac{RT}{P times mol. weight} right ) left (- frac{P times mol. weight}{R} frac{1}{T^2} right) = frac{1}{T}

Therefore, beta for an ideal gas is simply:

beta = frac{1}{T}

Thus, the Grashof number can be thought of as the ratio of the upwards buoyancy of the heated fluid to the internal friction slowing it down. In very sticky, viscous fluids, fluid movement is restricted, along with natural convection. In the extreme case of infinite viscosity, the fluid could not move and all heat transfer would be through conductive heat transfer.

A similar equation can be written 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.

Gr= frac{g beta Delta C L^3}{nu^2}

The relative magnitudes of the Grashof and Reynolds number determine which form of convection dominates, if frac{Gr}{Re^2} gg 1 forced convection may be neglected, whereas if frac{Gr}{Re^2} ll 1 natural convection may be neglected. If the ratio is approximately one both forced and natural convection need to be taken into account.

Natural convection is highly dependent on the geometry of the hot surface, various correlations exist in order to determine the heat transfer coefficent. The Rayleigh number (Ra ) is frequently used, where:

Ra = GrPr where Pr is the Prandtl number

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_4left(Prright)}{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:

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) 0.36pi

Forced convective heat transfer (advective heat transfer)

Forced convection is a mechanism, or type of heat transport in which fluid motion is generated by an external source (like a pump, fan, suction device, etc.). Forced convection is often encountered by engineers designing or analyzing heat exchangers, pipe flow, and flow over a plate at a different temperature than the stream (the case of a shuttle wing during re-entry, for example). However, in any forced convection situation, some amount of natural convection is always present. When the natural convection is not negligible, such flows are typically referred to as mixed convection.

When analysing potentially mixed convection, a parameter called the Archimedes number (Ar) parametizes the relative strength of free and forced convection. The Archimedes number is the ratio of Grashof number and the square of Reynolds number, which represents the ratio of buoyancy force and inertia force, and which stands in for the contribution of natural convection. When Ar >> 1, natural convection dominates and when Ar << 1, forced convection dominates.

Ar= frac{Gr}{Re^2}

When natural convection isn't a significant factor, mathematical analysis with forced convection theories typically yields accurate results. The parameter of importance in forced convection is the Peclet number, which is the ratio of advection (movement by currents) and diffusion (movement from high to low concentrations) of heat.

Pe=frac{U L}{alpha }

When the Peclet number is much greater than unity (1), advection dominates diffusion. Similarly, much smaller ratios indicate a higher rate of diffusion relative to advection.

References

  • Cebeci, Tuncer (2002). Convective Heat Transfer. Springer. ISBN 096684615X.
  • Burmeister, Louis (1993). Convective Heat Transfer, 2E. Wiley-Interscience. ISBN 047157709X.
  • Hewitt, G.F (1994). Process Heat Transfer. CRC Press Inc. ISBN 0-8493-9918-1.

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