# 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:

Q_{h} |
= | heat flow across the source boundary (M/T^{3}), e.g., W m^{−2} |

k_{T} |
= | thermal conductivity (M/(LΘ)), e.g., W m^{−1}K^{−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.