# Nusselt number

In thermal fluid dynamics, the **Nusselt number** (**Nu**, after Wilhelm Nusselt[1]^{: 336 }) is the ratio of convective to conductive heat transfer at a boundary in a fluid. 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. It is a dimensionless number, closely related to the fluid's Rayleigh number.[1]^{: 466 }

A Nusselt number of value one (zero) represents heat transfer by pure conduction.[1]^{: 336 } A value between one (zero) and 10 is characteristic of slug flow or laminar flow.[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. The mass transfer analogue of the Nusselt number is the Sherwood number.

## Definition

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.

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 (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*, the local Nusselt number is defined by taking the length to be the distance from the surface boundary[1] 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]

## Context

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.

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:

- , where
*q*is the heat transfer rate,*k*is the constant thermal conductivity and*T*the fluid temperature.

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 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[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 }

**Forced convection on flat plate**

### 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[4]^{: 490 }

The average Nusselt number for laminar flow over a flat plate, from the edge of the plate to a downstream distance , is given by[4]^{: 490 }

### Sphere in convective flow

In some applications, such as the evaporation of spherical liquid droplets in air, the following correlation is used:[5]

#### Gnielinski correlation

Gnielinski's correlation for turbulent flow in tubes:[4]^{: 490, 515 }[6]

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:[4]^{: 490 }

The Gnielinski Correlation is valid for:[4]^{: 490 }

#### Dittus–Boelter equation

The Dittus–Boelter equation (for turbulent flow) as introduced by W.H. McAdams[7] 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:

where:

- is the inside diameter of the circular duct
- is the Prandtl number
- for the fluid being heated, and for the fluid being cooled.[4]
^{: 493 }

The Dittus–Boelter equation is valid for[4]^{: 514 }

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 (68 °F), viscosity 10.07×10^{−4} Pa.s and a heat transfer surface temperature of 40 °C (104 °F) (viscosity 6.96×10^{−4} Pa.s, 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 (212 °F) (viscosity 2.82×10^{−4} Pa.s), 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 ( 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.[8]

- [4]
^{: 493 }

where:

- is the fluid viscosity at the bulk fluid temperature
- is the fluid viscosity at the heat-transfer boundary surface temperature

The Sieder–Tate correlation is valid for[4]^{: 493 }

### 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:

*D*= Hydraulic diameter_{h}*k*= thermal conductivity of the fluid_{f}*h*= convective heat transfer coefficient

#### Convection with uniform temperature for circular tubes

From Incropera & DeWitt,[4]^{: 486–487 }

OEIS sequence A282581 gives this value as .

#### Convection with uniform heat flux for circular tubes

For the case of constant surface heat flux,[4]^{: 486–487 }

## See also

- Sherwood number (mass transfer Nusselt number)
- Churchill–Bernstein equation
- Biot number
- Reynolds number
- Convective heat transfer
- Heat transfer coefficient
- Thermal conductivity

## References

- Çengel, Yunus A. (2002).
*Heat and Mass Transfer*(2nd ed.). McGraw-Hill. - "The Nusselt Number".
*Whiting School of Engineering*. Retrieved 3 April 2019. - 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. - Incropera, Frank P.; DeWitt, David P. (2007).
*Fundamentals of Heat and Mass Transfer*(6th ed.). Hoboken: Wiley. ISBN 978-0-471-45728-2. - McAllister, Sara; Chen, Jyh-Yuan; Fernández Pello, Carlos (2011). "Droplet Vaporization in Convective Flow".
*Fundamentals of combustion processes*. Mechanical Engineering. New York: Springer. p. 159. doi:10.1007/978-1-4419-7943-8. ISBN 978-1-4419-7942-1. LCCN 2011925371. - Gnielinski, Volker (1975). "Neue Gleichungen für den Wärme- und den Stoffübergang in turbulent durchströmten Rohren und Kanälen".
*Forsch. Ing.-Wes*.**41**(1): 8–16. doi:10.1007/BF02559682. S2CID 124105274. - Winterton, R.H.S. (February 1998). "Where did the Dittus and Boelter equation come from?" (PDF).
*International Journal of Heat and Mass Transfer*. Elsevier.**41**(4–5): 809–810. doi:10.1016/S0017-9310(97)00177-4. - "Temperature Profile in Steam Generator Tube Metal" (PDF). Archived from the original (PDF) on 3 March 2016. Retrieved 23 September 2009.

## External links

- Simple derivation of the Nusselt number from Newton's law of cooling (Accessed 23 September 2009)