Chapter 2 - In depth

# Regime 5: Natural convection

In determining convective heat transfer coefficients, typically empirical relations are used. These relations usually result in a value for the Nusselt number, which can be used to calculate the heat transfer coefficient.  There are many relations, and selecting the relevant one depends on correct determination of the convective flow regime and geometry. Here, an overview is given of a selection of general relations. Somewhat more accurate relations can typically be found for more specific geometries or flow regimes. Typically, the accuracy of the relations given here is better than 10%.

To calculate a connective heat transfer coefficient, take the following steps:

1. Select the relevant configuration in the table below
2. Use the given correlation and the dimensionless numbers definition list to find a value for the Nusselt number. Note: a dimensionless number with a subscript denotes it is calculated with a specific relevant parameter, for example Re_D is the Reynolds number calculated using the diameter D as the length scale.
3. Finally, use the calculated value and the definition of Nu to find the heat transfer coefficient:

$$h = \frac{k}{L} Nu$$

## Immersed body

A very general formula covering most immersed body shapes is:

$$Nu_{L} = 3.47 + 0.51 Ra_{L}^{1/4}$$

Here, the length parameter used in calculating Nu and Ra is the square root of the total immersed surface area:

$$L = A^{1/2}$$

## Isothermal vertical wall

This relation is wall for a vertical wall kept at a constant temperature. Convective heat transfer along a vertical wall is complex, due to interaction of buoyancy and inertial forces, transition to turbulence and a varying Nusselt number along the height of the wall. Nevertheless, Churchill and Chu have developed an empirical correlation for the height-averaged Nusselt number over wide range of regimes (laminar through turbulent):

$$\overline{{Nu}_{y}} = \left\{0.825 + \frac{0.387 Ra_{y}^{1/6}}{[1 + (0.492/Pr)^{9/16}]^{8/27}} \right\}^{2}$$

## Constant heat flux vertical wall

Very similar to the above relation, this one is valid for the case where a constant heat flux is transferred from the wall to the fluid. For example, a situation where the wall is uniformly heated at constant power.  Since this means that the wall temperature is not uniform, the y-averaged wall temperature is used in the Rayleigh formula. If the heat flux is known and the wall temperature is unknown, a bit of algebra is required to solve for the average wall temperature.

$$\overline{{Nu}_{y}} = \left\{0.825 + \frac{0.387 Ra_{y}^{1/6}}{[1 + (0.437/Pr)^{9/16}]^{8/27}} \right\}^{2}$$

## Inclined wall

For an inclined wall, the above relations for vertical walls can be used by modifying the gravity term in the Rayleigh number.  Simply replace the gravitational acceleration constant g by cos(φ)*g to account for the slope. Here φ=0 degrees is a vertical wall. The approximation is valid up to around φ=60 degrees.

## Horizontal surface

Free convection at a horizontal surface is dependent on the direction vertical motion of the convective flow.

For a hot surface facing upward, or a cold surface facing downward:

$$\overline{{Nu_{L}}} = 0.54 Ra_{L}^{1/4} \qquad (10^{4} < Ra < 10^{7})$$

$$\overline{{{Nu}_{L}}} = 0.15 Ra_{L}^{1/3} \qquad (10^{7} < Ra < 10^{9})$$

For a hot surface facing downward, or a cold surface facing upward:

$$\overline{{Nu}_{L}} = 0.27 Ra_{L}^{1/4} \qquad (10^{5} < Ra < 10^{10})$$

In these correlations, the characteristic length parameter used to compute Ra is the area divided by the perimeter of the surface:

$$L = \frac{A}{p}$$

## Horizontal isothermal cylinder

The correlation for a horizontal cylinder looks very similar to that of a vertical wall. It is given by:

$$\overline{{Nu}_{D}} = \left\{0.6 + \frac{0.387 Ra_{D}^{1/6}}{[1 + (0.559/Pr)^{9/16}]^{8/27}} \right\}^{2}$$

Here, the length scales are based on the cylinder diameter D.

## Vertical isothermal cylinder

Free convective heat transfer from a vertical cylinder of height H and diameter D depends on its aspect ratio and involves three surfaces (cylinder and top and bottom discs). Therefore, the relation contains two terms and includes the aspect ratio H/D.

$$\overline{{Nu}_{H}} = \frac{4}{3} \left( \frac{7 Ra_{H} Pr}{5 (20 + 21 Pr)} \right)^{1/4} + \frac{4 (272 + 315 Pr) H}{35 (64 + 63Pr) D}$$

## Vertical channel (Chimney)

Free convective heat flow in a narrow vertical channel with isothermal walls (“chimney flow”) is dependent on the channel cross section.  We consider a “narrow” channel only, since in a “wide” channel the walls do not affect each other. In that case, simply use the vertical wall correlation to consider the heat transfer at each wall separately.  The criterion for a “narrow” channel is:

$$Ra_{D_H} \gt H/D_{H}$$

Here, H is the vertical height of the channel, and DH is the hydraulic diameter of its cross-section, defined as 4 times the area-perimeter ratio:

$$D_{H} = 4 A/p$$

A simple empirical relation for the Nusselt number is based on this hydraulic diameter:

$$\overline{{Nu}_{H}} = Ra_{D_H}/c$$

Here, c is a parameter dependent on the cross-sectional shape:

## Horizontal gap

A gap is formed between two parallel horizontal plates, a distance H apart.

No natural convection will occur if the upper plate is warmer than the lower plate. The fluid in between will stratify a form a linear temperature gradient. Heat transfer occurs only via conduction through the static medium.

If the lower plate is warmer than the upper plate, an unstable situation might arise. At Rayleigh number below a critical value of 1708, no convection occurs, and pure heat conduction occurs along a linear temperature gradient.

For RaH>1708, the convection sets in. Recirculating convection rolls will appear, transferring heat from the bottom to the top plate. In this regime, the following heat transfer correlation is given:

$$Nu_{H} = 0.069 Ra_{L}^{1/3} Pr^{0.074}$$

Note that here, we are considering the net heat transfer from one plate to the other, whereas in the other configurations, we consider the heat transfer between the solid wall and the fluid itself.