Nusselt number

 ( Convection ) 

In Thermal fluids, the Nusselt number (, after Wilhelm Nusselt) is the ratio of total heat transfer to heat conduction 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. 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.

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Definition The Nusselt number is the ratio of total heat transfer (convection + conduction) 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.

$\backslash mathrm\{Nu\}\_L\; =\; \backslash frac\{\backslash mbox\{Total\; heat\; transfer\; \}\}\{\backslash mbox\{Conductive\; heat\; transfer\; \}\}\; =\; \backslash frac\{h\}\{k/L\}\; =\; \backslash frac\{hL\}\{k\}$

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 to the local point of interest.

$\backslash mathrm\{Nu\}\_x\; =\; \backslash frac\{h\_x\; x\}\{k\}$

The mean, or average, number is obtained by integrating the expression over the range of interest, such as:

$\backslash overline\{\backslash mathrm\{Nu\}\}=\backslash frac\{\backslash frac\{1\}\{L\}\; \backslash int\_0^L\; h\_x\backslash \; dx\backslash \; L\}\{k\}=\backslash frac\{\backslash overline\{h\}\; L\}\{k\}$

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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.

The heat transfer rate can be written using Newton's law of cooling as

$Q\_y=hA\backslash left(\; T\_s-T\_\backslash infty\; \backslash right)$,

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:

$Q\_y=-kA\backslash frac\{\backslash partial\; \}\{\backslash partial\; y\}\backslash left.\; \backslash left(\; T-T\_s\; \backslash right)\; \backslash right|\_\{y=0\}$.

These two terms are equal; thus

$-kA\backslash frac\{\backslash partial\; \}\{\backslash partial\; y\}\backslash left.\; \backslash left(\; T-T\_s\; \backslash right)\; \backslash right|\_\{y=0\}=hA\backslash left(\; T\_s-T\_\backslash infty\; \backslash right)$.

Rearranging,

$\backslash frac\{h\}\{k\}=\backslash frac\{\backslash left.\; \backslash frac\{\backslash partial\; \backslash left(\; T\_s-T\; \backslash right)\}\{\backslash partial\; y\}\; \backslash right|\_\{y=0\}\}\{\backslash left(\; T\_s-T\_\backslash infty\; \backslash right)\}$.

Multiplying by a representative length L gives a dimensionless expression:

$\backslash frac\{hL\}\{k\}=\backslash frac\{\backslash left.\; \backslash frac\{\backslash partial\; \backslash left(\; T\_s-T\; \backslash right)\}\{\backslash partial\; y\}\; \backslash right|\_\{y=0\}\}\{\backslash frac\{\backslash left(\; T\_s-T\_\backslash infty\; \backslash right)\}\{L\}\}$.

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.

$\backslash mathrm\{Nu\}\; =\; \backslash frac\{h\}\{k/L\}\; =\; \backslash frac\{hL\}\{k\}$.

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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:

$q\; =\; -k\; A\; \backslash nabla\; T$, where q is the heat transfer rate, k is the constant thermal conductivity and T the fluid temperature.

Indeed, if: $\backslash nabla\text{'}\; =\; L\; \backslash nabla$ and $T\text{'}\; =\; \backslash frac\{T-T\_h\}\{T\_h-T\_c\}$

we arrive at

$-\backslash nabla\text{'}T\text{'}\; =\; \backslash frac\{L\}\{kA(T\_h-T\_c)\}q=\backslash frac\{hL\}\{k\}$

then we define

$\backslash mathrm\{Nu\}\_L=\backslash frac\{hL\}\{k\}$

so the equation becomes

$\backslash mathrm\{Nu\}\_L=-\backslash nabla\text{'}T\text{'}$

By integrating over the surface of the body:

$\backslash overline\{\backslash mathrm\{Nu\}\}=-\; \backslash int\_\{S\text{'}\}^\{\}\; \backslash mathrm\{Nu\}\; \backslash ,\; \backslash mathrm\{d\}S\text{'}\backslash !$,

where $S\text{'}\; =\; \backslash frac\{S\}\{L^2\}$.

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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:

$\backslash mathrm\{Nu\}\; =\; f(\backslash mathrm\{Ra\},\; \backslash mathrm\{Pr\})$

Otherwise, for forced convection, the Nusselt number is generally a function of the Reynolds number and the Prandtl number, or

$\backslash mathrm\{Nu\}\; =\; f(\backslash mathrm\{Re\},\; \backslash mathrm\{Pr\})$

correlations for a wide variety of geometries are available that express the Nusselt number in the aforementioned forms.

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Free convection

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Free convection at a vertical wall Cited as coming from Churchill and Chu:

$\backslash overline\{\backslash mathrm\{Nu\}\}\_L\; \backslash \; =\; 0.68\; +\; \backslash frac\{0.663\backslash ,\; \backslash mathrm\{Ra\}\_L^\{1/4\}\}\{\backslash left1^\{4/9\}\; \backslash ,\}\; \backslash quad\; \backslash mathrm\{Ra\}\_L\; \backslash le\; 10^8$

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Free convection from horizontal plates If the characteristic length is defined

$L\; \backslash \; =\; \backslash frac\{A\_s\}\{P\}$

where $\backslash mathrm\{A\}\_s$ is the surface area of the plate and $P$ 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

$\backslash overline\{\backslash mathrm\{Nu\}\}\_L\; \backslash \; =\; 0.54\backslash ,\; \backslash mathrm\{Ra\}\_L^\{1/4\}\; \backslash ,\; \backslash quad\; 10^4\; \backslash le\; \backslash mathrm\{Ra\}\_L\; \backslash le\; 10^7$

$\backslash overline\{\backslash mathrm\{Nu\}\}\_L\; \backslash \; =\; 0.15\backslash ,\; \backslash mathrm\{Ra\}\_L^\{1/3\}\; \backslash ,\; \backslash quad\; 10^7\; \backslash le\; \backslash mathrm\{Ra\}\_L\; \backslash le\; 10^\{11\}$

And for the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment

$\backslash overline\{\backslash mathrm\{Nu\}\}\_L\; \backslash \; =\; 0.52\backslash ,\; \backslash mathrm\{Ra\}\_L^\{1/5\}\; \backslash ,\; \backslash quad\; 10^5\; \backslash le\; \backslash mathrm\{Ra\}\_L\; \backslash le\; 10^\{10\}$

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Free convection from enclosure heated from below Cited

(2025). 9780470900376, Wiley. . ISBN 9780470900376

as coming from Bejan:

$\backslash overline\{\backslash mathrm\{Nu\}\}\_L\; \backslash \; =\; 0.069\backslash ,\; \backslash mathrm\{Ra\}\_L^\{1/3\}Pr^\{0.074\}\; \backslash ,\; \backslash quad\; 3\; *\; 10^5\; \backslash le\; \backslash mathrm\{Ra\}\_L\; \backslash le\; 7\; *\; 10^\{9\}$

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.

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Flat plate in laminar flow The local Nusselt number for laminar flow over a flat plate, at a distance $x$ downstream from the edge of the plate, is given by

$\backslash mathrm\{Nu\}\_x\backslash \; =\; 0.332\backslash ,\; \backslash mathrm\{Re\}\_x^\{1/2\}\backslash ,\; \backslash mathrm\{Pr\}^\{1/3\},\; (\backslash mathrm\{Pr\}\; >\; 0.6)$

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

$\backslash overline\{\backslash mathrm\{Nu\}\}\_x\; \backslash \; =\; \{2\}\; \backslash cdot\; 0.332\backslash ,\; \backslash mathrm\{Re\}\_x^\{1/2\}\backslash ,\; \backslash mathrm\{Pr\}^\{1/3\}\backslash \; =\; 0.664\backslash ,\; \backslash mathrm\{Re\}\_x^\{1/2\}\backslash ,\; \backslash mathrm\{Pr\}^\{1/3\},\; (\backslash mathrm\{Pr\}\; >\; 0.6)$

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

(2025). 9781441979421, Springer. ISBN 9781441979421

$\backslash mathrm\{Nu\}\_D\; \backslash \; =\; \{2\}\; +\; 0.4\backslash ,\; \backslash mathrm\{Re\}\_D^\{1/2\}\backslash ,\; \backslash mathrm\{Pr\}^\{1/3\}\backslash ,$

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Forced convection in turbulent pipe flow

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Gnielinski correlation Gnielinski's correlation for turbulent flow in tubes:

(2025). 9780471457282, Wiley. . ISBN 9780471457282

$\backslash mathrm\{Nu\}\_D\; =\; \backslash frac\{\; \backslash left(\; f/8\; \backslash right)\; \backslash left(\; \backslash mathrm\{Re\}\_D\; -\; 1000\; \backslash right)\; \backslash mathrm\{Pr\}\; \}\; \{1\; +\; 12.7(f/8)^\{1/2\}\; \backslash left(\; \backslash mathrm\{Pr\}^\{2/3\}\; -\; 1\; \backslash right)\}$

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:

$f=\; \backslash left(\; 0.79\; \backslash ln\; \backslash left(\backslash mathrm\{Re\}\_D\; \backslash right)-1.64\; \backslash right)^\{-2\}$

The Gnielinski Correlation is valid for:

$0.5\; \backslash le\; \backslash mathrm\{Pr\}\; \backslash le\; 2000$

$3000\; \backslash le\; \backslash mathrm\{Re\}\_D\; \backslash le\; 5\; \backslash times\; 10^\{6\}$

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Dittus–Boelter equation The Dittus–Boelter equation (for turbulent flow) 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 (most commercial applications) is cautioned. The Dittus–Boelter equation is:

$\backslash mathrm\{Nu\}\_D\; =\; 0.023\backslash ,\; \backslash mathrm\{Re\}\_D^\{4/5\}\backslash ,\; \backslash mathrm\{Pr\}^\{n\}$

where:

$D$ is the inside diameter of the circular duct

$\backslash mathrm\{Pr\}$ is the Prandtl number

$n\; =\; 0.4$ for the fluid being heated, and $n\; =\; 0.3$ for the fluid being cooled.

The Dittus–Boelter equation is valid for

$0.6\; \backslash le\; \backslash mathrm\{Pr\}\; \backslash le\; 160$

$\backslash mathrm\{Re\}\_D\; \backslash gtrsim\; 10\backslash ,000$

$\backslash frac\{L\}\{D\}\; \backslash gtrsim\; 10$

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 (viscosity , a viscosity correction factor for $(\{\backslash mu\}\; /\; \{\backslash mu\_s\})$ can be obtained as 1.45. This increases to 3.57 with a heat transfer surface temperature of (viscosity ), making a significant difference to the Nusselt number and the heat transfer coefficient.

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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 ($\backslash mu$ and $\backslash mu\_s$) 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.

$\backslash mathrm\{Nu\}\_D\; =\; 0.027\backslash ,\backslash mathrm\{Re\}\_D^\{4/5\}\backslash ,\; \backslash mathrm\{Pr\}^\{1/3\}\backslash left(\backslash frac\{\backslash mu\}\{\backslash mu\_s\}\backslash right)^\{0.14\}$

where:

$\backslash mu$ is the fluid viscosity at the bulk fluid temperature

$\backslash mu\_s$ is the fluid viscosity at the heat-transfer boundary surface temperature

The Sieder–Tate correlation is valid for

$0.7\; \backslash le\; \backslash mathrm\{Pr\}\; \backslash le\; 16\backslash ,700$

$\backslash mathrm\{Re\}\_D\; \backslash ge\; 10\backslash ,000$

$\backslash frac\{L\}\{D\}\; \backslash gtrsim\; 10$

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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:

$\backslash mathrm\{Nu\}\; =\; \backslash frac\{h\; D\_h\}\{k\_f\}$

where:

D_h = Hydraulic diameter

k_f = thermal conductivity of the fluid

h = convective heat transfer coefficient

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Convection with uniform temperature for circular tubes From Incropera & DeWitt,

$\backslash mathrm\{Nu\}\_D\; =\; 3.66$

OEIS sequence gives this value as $\backslash mathrm\{Nu\}\_D\; =\; 3.6567934577632923619...$.

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Convection with uniform heat flux for circular tubes For the case of constant surface heat flux,

$\backslash mathrm\{Nu\}\_D\; =\; 4.36$

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See also

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

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External links

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

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