Nusselt Number

In heat transfer at a boundary (surface) within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across (normal to) the boundary. In this context, convection includes both advection and conduction. Named after Wilhelm Nusselt, it is a dimensionless number. The conductive component is measured under the same conditions as the heat convection but with a (hypothetically) stagnant (or motionless) fluid.

A Nusselt number close to one, namely convection and conduction of similar magnitude, is characteristic of "slug flow" or laminar flow. A larger Nusselt number corresponds to more active convection, with turbulent flow typically in the 100–1000 range.

The convection and conduction heat flows are parallel to each other and to the surface normal of the boundary surface, and are all perpendicular to the mean fluid flow in the simple case.

where:

• L = characteristic length
• k_f = thermal conductivity of the fluid
• h = convective heat transfer coefficient

Selection of the characteristic length should be in the direction of growth (or thickness) of the boundary layer. Some examples of characteristic length are: the outer diameter of a cylinder in (external) cross flow (perpendicular to the cylinder axis), the length of a vertical plate undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may be defined as the volume of the fluid body divided by the surface area. The thermal conductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineering purposes may be calculated as the mean-average of the bulk fluid temperature and wall surface temperature. For relations defined as a local Nusselt number, one should take the characteristic length to be the distance from the surface boundary to the local point of interest. However, to obtain an average Nusselt number, one must integrate said relation over the entire characteristic length.

Typically, for free convection, the average Nusselt number is expressed as a function of the Rayleigh number and the Prandtl number, written as: Nu = f(Ra, Pr). Else, for forced convection, the Nusselt number is generally a function of the Reynolds number and the Prandtl number, or Nu = f(Re, Pr). Empirical correlations for a wide variety of geometries are available that express the Nusselt number in the aforementioned forms.

The mass transfer analog of the Nusselt number is the Sherwood number.