Nusselt number

Nusselt number
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In heat transfer at a boundary (surface) within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across (normal to) the boundary. Named after Wilhelm Nusselt, it is a dimensionless number. The conductive component is measured under the same conditions as the heat convection but with a (hypothetically) stagnant (or motionless) fluid.

A Nusselt number close to unity, namely convection and conduction of similar magnitude, 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.

¹. 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.

$\mathit{Nu}_L = \frac{hL}{k_f} = \frac{\mbox{Convective heat transfer}}{\mbox{Conductive heat transfer}}$

where:

L = characteristic length
k_f = thermal conductivity of the fluid
h = convective heat transfer coefficient

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. For relations defined as a local Nusselt number, one should take the characteristic length to be the distance from the surface boundary to the local point of interest. However, to obtain an average Nusselt number, one must integrate said relation over the entire characteristic length.

Typically the average Nusselt number is expressed as a function of the Rayleigh number and the Prandtl number, written as: Nu = f(Ra, Pr). Empirical correlations for a wide variety of geometries are available that express the Nusselt number in the aforementioned form.

The mass transfer analog of the Nusselt number is the Sherwood number.

content

1 Empirical calculations
2 See also
3 References

Empirical calculations

Free convection at a vertical wall

Cited as coming from Churchill and Chu²

$\overline{Nu}_L \ = 0.68 + \frac{0.67Ra_L^{1/4}}{\left[1 + (0.492/Pr)^{9/16} \, \right]^{4/9} \,} \quad Ra_L \le 10^9$

Free convection from horizontal plates

For the top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment²

$\overline{Nu}_L \ = 0.54 Ra_L^{1/4} \, \quad 10^4 \le Ra_L \le 10^7$

$\overline{Nu}_L \ = 0.15 Ra_L^{1/3} \, \quad 10^7 \le Ra_L \le 10^{11}$

For the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment²

$\overline{Nu}_L \ = 0.27 Ra_L^{1/4} \, \quad 10^5 \le Ra_L \le 10^{10}$

Forced convection in turbulent pipe flow

Dittus-Boelter equation

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. The Dittus-Boelter equation is:

$Nu_D = 0.023 Re_D^{4/5} Pr^{n}$

where:

D refers to the "characteristic length" used to calculate the respective parameter
Pr is the Prandtl number
n=0.4 for heating of the fluid, and n=0.3 for cooling of the fluid²

The following is an example of how 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 degrees Celsius, (viscosity = 10.07*10^-4) and a heat transfer surface temperature of 40 degrees Celsius (viscosity = 6.96*10^-4), a viscosity correction factor for $(u / u s)$ can be obtained as 1.45. This increases to 3.57 with a heat transfer surface temperature of 100 degrees Celsius (viscosity =2.82*10^-4), making a significant difference to the Nusselt number and the heat transfer coefficient.

Sieder-Tate correlation

The Sieder-Tate correlation (also for turbulent flow) is an implicit function, as it analyses 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 ( $u, u s$ ) due to temperature change between the bulk fluid average temperature and the heat transfer (s)urface temperature, respectively. The Sieder-Tate correlation is normally solved by an iterative process, as the viscosity term will change as the Nusselt number changes.³

$Nu_D = 0.027 Re_D^{4/5} Pr^{1/3}(u/u_s)^{0.14}$

where:

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

Forced convection in laminar fully developed pipe flow

For fully developed laminar flow, the Nusselt numbers are much simpler.

Convection with uniform surface heat flux

⁴

$N u D = 4.36$

Convection with uniform surface temperature

$N u D = 3.66$

References

^ Simple derivation of the Nusselt number from Newton's law of cooling
^ ^a ^b ^c ^d Incropera, Frank P.; DeWitt, David P. Fundamentals of Heat and Mass Transfer, 4th Edition, Wiley, p. 493.
^ 24.217 Computer Applications
^ Incropera, Frank P.; DeWitt, David P. Fundamentals of Heat and Mass Transfer, 5th Edition, Wiley, pp. 486, 487.