Nusselt number
In thermal fluid dynamics, the Nusselt number (Nu, after Wilhelm Nusselt[1]: 336 ) is the ratio of total heat transfer to conductive heat transfer at a boundary in a fluid.
Total heat transfer combines conduction and convection.
Convection includes both advection (fluid motion) and diffusion (conduction).
The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid.
[1]: 466 A Nusselt number of order one represents heat transfer by pure conduction.
[2] A larger Nusselt number corresponds to more active convection, with turbulent flow typically in the 100–1000 range.
[2] A similar non-dimensional property is the Biot number, which concerns thermal conductivity for a solid body rather than a fluid.
where h is the convective heat transfer coefficient of the flow, L is the characteristic length, and k is the thermal conductivity of the fluid.
In contrast to the definition given above, known as average Nusselt number, the local Nusselt number is defined by taking the length to be the distance from the surface boundary[1][page needed] to the local point of interest.
The mean, or average, number is obtained by integrating the expression over the range of interest, such as:[3] An understanding of convection boundary layers is necessary to understand convective heat transfer between a surface and a fluid flowing past it.
Because heat transfer at the surface is by conduction, the same quantity can be expressed in terms of the thermal conductivity k: These two terms are equal; thus Rearranging, Multiplying by a representative length L gives a dimensionless expression: 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.
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: Indeed, if:
Cited[4]: 493 as coming from Churchill and Chu: If the characteristic length is defined where
Then for the top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment[4]: 493 And for the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment[4]: 493 Cited[5] as coming from Bejan: This equation "holds when the horizontal layer is sufficiently wide so that the effect of the short vertical sides is minimal."
It was empirically determined by Globe and Dropkin in 1959:[6] "Tests were made in cylindrical containers having copper tops and bottoms and insulating walls."
The local Nusselt number for laminar flow over a flat plate, at a distance
McAdams[9] 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.
The Sieder–Tate correlation for turbulent flow is an implicit function, as it analyzes the system as a nonlinear boundary value problem.
[10] where: The Sieder–Tate correlation is valid for[4]: 493 For fully developed internal laminar flow, the Nusselt numbers tend towards a constant value for long pipes.
