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A Nusselt number of value one represents heat transfer by pure conduction. A value between one and 10 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 Nusselt number is named after Wilhelm Nusselt, who made significant contributions to the science of convective heat transfer.
The Nusselt number is the ratio of convective to conductive heat transfer across a boundary. 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.
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.
In contrast to the definition given above, known as average Nusselt number, local Nusselt number is defined by taking the length to be the distance from the surface boundary to the local point of interest.
The mean, or average, number is obtained by integrating the expression over the range of interest, such as:
An understanding of convection boundary layers is necessary to understanding convective heat transfer between a surface and a fluid flowing past it. A thermal boundary layer develops if the fluid free stream temperature and the surface temperatures differ. A temperature profile exists due to the energy exchange resulting from this temperature difference.
Thermal Boundary Layer
The heat transfer rate can then be written as,
And because heat transfer at the surface is by conduction,
These two terms are equal; thus
Making it dimensionless by multiplying by representative length L,
The right hand side is now the ratio of the temperature gradient at the surface to the reference temperature gradient, while the left hand side is similar to the Biot modulus. This becomes the ratio of conductive thermal resistance to the convective thermal resistance of the fluid, otherwise known as the Nusselt number, Nu.
The Nusselt number may be obtained by a non-dimensional analysis of Fourier's law since it is equal to the dimensionless temperature gradient at the surface:
The Dittus-Boelter equation (for turbulent flow) is an explicit function for calculating the Nusselt number. It is easy to solve but is less accurate when there is a large temperature difference across the fluid. It is tailored to smooth tubes, so use for rough tubes (most commercial applications) is cautioned. The Dittus-Boelter equation is:
Example The Dittus-Boelter equation is a good approximation where temperature differences between bulk fluid and heat transfer surface are minimal, avoiding equation complexity and iterative solving. Taking water with a bulk fluid average temperature of 20 °C, viscosity 10.07×10−4 Pa·s and a heat transfer surface temperature of 40 °C (viscosity 6.96×10−4, a viscosity correction factor for can be obtained as 1.45. This increases to 3.57 with a heat transfer surface temperature of 100 °C (viscosity 2.82×10−4 Pa·s), making a significant difference to the Nusselt number and the heat transfer coefficient.
The Sieder-Tate correlation for turbulent flow is an implicit function, as it analyzes the system as a nonlinear boundary value problem. The Sieder-Tate result can be more accurate as it takes into account the change in viscosity ( and ) due to temperature change between the bulk fluid average temperature and the heat transfer surface temperature, respectively. The Sieder-Tate correlation is normally solved by an iterative process, as the viscosity factor will change as the Nusselt number changes.
^Çengel, Yunus A. (2002). Heat and Mass Transfer (Second ed.). McGraw-Hill. p. 336.
^Yunus A. Çengel (2003). Heat Transfer: a Practical Approach (2nd ed.). McGraw-Hill.
^E. Sanvicente; et al. (2012). "Transitional natural convection flow and heat transfer in an open channel". International Journal of Thermal Sciences. 63: 87-104. doi:10.1016/j.ijthermalsci.2012.07.004.