Nusselt number

In thermal fluid dynamics, the Nusselt number 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 and diffusion. The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid. It is a dimensionless number, closely related to the fluid's Rayleigh number.
A Nusselt number of order 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.
A similar non-dimensional property is the Biot number, which concerns thermal conductivity for a solid body rather than a fluid. The mass transfer analogue of the Nusselt number is the Sherwood number.

Definition

The Nusselt number is the ratio of total heat transfer 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.
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.

Selection of the characteristic length should be in the direction of growth of the boundary layer; some examples of characteristic length are: the outer diameter of a cylinder in cross flow, 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 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, the 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:

Context

An understanding of convection boundary layers is necessary to understand 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.
[Image:Thermal Boundary Layer.jpg|400px|thumb|Thermal Boundary Layer]
The heat transfer rate can be written using Newton's law of cooling as
where h is the heat transfer coefficient and A is the heat transfer surface area. 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. This becomes the ratio of conductive thermal resistance to the convective thermal resistance of the fluid, otherwise known as the Nusselt number, Nu.

Derivation

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: and
we arrive at
then we define
so the equation becomes
By integrating over the surface of the body:
where.

Empirical correlations

Typically, for free convection, the average Nusselt number is expressed as a function of the Rayleigh number and the Prandtl number, written as:
Otherwise, for forced convection, the Nusselt number is generally a function of the Reynolds number and the Prandtl number, or
Empirical correlations for a wide variety of geometries are available that express the Nusselt number in the aforementioned forms.

Free convection

Free convection at a vertical wall

Cited as coming from Churchill and Chu:

Free convection from horizontal plates

If the characteristic length is defined
where is the surface area of the plate and is its perimeter.
Then for the top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment
And for the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment

Free convection from enclosure heated from below

Cited 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: "Tests were made in cylindrical containers having copper tops and bottoms and insulating walls." The containers used were around 5" in diameter and 2" high.

Flat plate in laminar flow

The local Nusselt number for laminar flow over a flat plate, at a distance downstream from the edge of the plate, is given by
The average Nusselt number for laminar flow over a flat plate, from the edge of the plate to a downstream distance, is given by

Sphere in convective flow

In some applications, such as the evaporation of spherical liquid droplets in air, the following correlation is used:

Forced convection in turbulent pipe flow

Gnielinski correlation

Gnielinski's correlation for flow in tubes with :
where f is the Darcy friction factor that can either be obtained from the Moody chart or for smooth tubes from correlation developed by Petukhov:
The Gnielinski Correlation is valid for:

Dittus–Boelter equation

The Dittus–Boelter equation as introduced by W.H. McAdams 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 is cautioned. The Dittus–Boelter equation is:
where:
The Dittus–Boelter equation is valid for
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, viscosity and a heat transfer surface temperature of , making a significant difference to the Nusselt number and the heat transfer coefficient.

Sieder–Tate correlation

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 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.
where:
The Sieder–Tate correlation is valid for

Forced convection in fully developed laminar pipe flow

For fully developed internal laminar flow, the Nusselt numbers tend towards a constant value for long pipes.
For internal flow:
where:

Convection with uniform temperature for circular tubes

From Incropera & DeWitt,
OEIS sequence gives this value as.

Convection with uniform heat flux for circular tubes

For the case of constant surface heat flux,