#### Free Convection: Screencast

Explains the physics of free convection and introduces pertinent correlations.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

##### Important Equations:

Often the goal of problems involving free convection is to calculate the total heat flow (in Watts). You can do so using the following equation:

\[q = \bar{h} A(T_S \, -\, T_{\infty})\]

where \(\bar{h}\) = average heat transfer coefficient (W/m^{2}-K),

\(A\) = surface area of the object (m^{2}),

\(T_S\) = surface temperature of the object (°C),

and \(T_{\infty}\) = temperature of the fluid far from the object (°C).

To calculate \(\bar{h}\), you need to use a correlation for free convection. Two example correlations are:

\(\overline{Nu}_L = \left( 0.825\,+\, \frac{0.387Ra_L ^{1/6}}{[1\,+\,(0.492/Pr)^{9/16}]^{8/27}} \right)^2\) and \(\overline{Nu}_D = 2 \,+\, \frac{0.589Ra_D ^{1/4}}{[1+(0.469/Pr)^{9/16}]^{4/9}}\)

The first correlation is for a vertical heated plate, with height \(L\). The second correlation is for a heated sphere with diameter \(D\). The Nusselt number is defined as:

\(\overline{Nu}_L = \frac{\bar{h} L}{k}\) and \(\overline{Nu}_D = \frac{\bar{h} D}{k}\)

where \(k\) = thermal conductivity of the fluid (W/m^{2}-K).

The Rayleigh number is defined by

\(Ra_L = Gr_L Pr = \frac{g\beta (T_S\,-\,T_{\infty}) L^3}{\nu \alpha}\)

where \(g\) = acceleration due to gravity (m^{2}/s^{2})

and \(\beta\) = volumetric thermal expansion coefficient (1/K). For an ideal gas, \(\beta = \frac{1}{T_{film}}\) where \(T_{film}\) must be in Kelvin (or degrees Rankine). If the fluid is not an ideal gas (water, for example), the you must look up its value of \(\beta\) in a table.

Because the temperature of the object and the temperature of the fluid are different, the usual approach is to use a “film” temperature when looking up the fluid properties. The film temperature is just the average of the two:

\(T_{film} = \frac{T_{\infty} \,+\, T_S}{2}\)

The Prandtl number is a property of the fluid that can be looked up for a given temperature, defined by \(Pr = \frac{\nu}{\alpha}\)

Note that the Rayleigh number is the product of the Prandtl number and the Grashof number, which is

\(Gr_L = \frac{g\beta (T_S \,-\,T_{\infty}) L^3}{\nu ^2}\)

Before using a given correlation, you must be certain that the relevant dimensionless numbers are within ranges appropriate for that correlation. For example, in the correlation provided above for spheres, \(Pr\) must be greater than or about equal to 0.7 and \(Ra_D\) must be less than or about equal to 10^{11}.

##### Important Definitions:

Kinematic viscosity: \(\nu = \frac{\mu}{\rho}\) with units of m^{2}/s

Thermal diffusivity: \(\alpha = \frac{k}{\rho c_p}\) with units of m^{2}/s