5.5 Nusselt Number
The Nusselt number (Nu) is equivalent to the Sherwood number but applies to heat transport. It is defined in Equation 31 (Elder, 1967a).
 |
(31) |
where:
| Qh | = | heat flow across the source boundary (M/T3), e.g., W m−2 |
| kT | = | thermal conductivity (M/(LΘ)), e.g., W m−1K−1 |
| ∆T | = | temperature difference over the height H (Θ), e.g., K |
Laboratory and computational experiments have shown that there is a relationship between the Rayleigh number and the Nusselt number (Cheng, 1979). This is illustrated in Figure 17. If the Rayleigh number is below the critical value, there is no convective heat transfer and so Nu = 1 (conductive heat transfer only). As the Rayleigh number increases, so does the Nusselt number, which means that more heat is being transported convectively. This is because at high Rayleigh numbers, the convective cells become slimmer and thereby more efficient in transferring heat. Modeling by Holzbecher (1996) showed that the exact relationship varies depending on the aspect ratio of the domain as illustrated in Figure 17, that is, the larger the height/width ratio, the higher the heat transfer at high Rayleigh numbers.
Figure 17 – Relationship between the Rayleigh (Ra) and Nusselt (Nu) numbers originally published by Cheng (1978). The lines represent model-based relationships by Holzbecher (1996) for domains of different height/width ratios H/W, as shown schematically to the right of the graph. The critical Rayleigh number for each H/W value is indicated by an arrow below the horizontal axis. Original figure reprinted from Advances in Heat Transfer, 14, P. Cheng, Heat Transfer in Geothermal Systems, Copyright (1979), with permission from Elsevier. Model data courtesy of Ekkehard Holzbecher.