Math 2.0

Classical thermodynamics, heat transfer Convective heat transfer Fact

Dittus–Boelter equation

Builds on Nusselt numberReynolds numberPrandtl numberForced convectionHeat transfer coefficient

Statement

Consider fully developed turbulent flow of a fluid in a smooth circular tube of inner diameter DD. Let NuD\mathrm{Nu}_D be the Nusselt number and ReD\mathrm{Re}_D the Reynolds number, both based on DD and the mean velocity. Let Pr⁡\Pr be the Prandtl number, with the fluid properties taken at the bulk mean temperature. Assume ReD≥104\mathrm{Re}_D \ge 10^4, 0.6≤Pr⁡≤1600.6 \le \Pr \le 160, tube length at least 10D10D, and a moderate difference between wall and fluid temperatures. Then, approximately, NuD=0.023 ReD4/5 Pr⁡n,\mathrm{Nu}_D = 0.023 \, \mathrm{Re}_D^{4/5} \, \Pr^{n}, with n=0.4n = 0.4 when the fluid is heated and n=0.3n = 0.3 when it is cooled.

Remarks

The equation is an empirical fit to measurements, not a consequence of a derivation. The exponent 4/54/5 agrees with the friction law for turbulent pipe flow through analogies between friction and heat transfer. Example: water with Pr⁡=5\Pr = 5 heated in a tube at ReD=5×104\mathrm{Re}_D = 5 \times 10^4 gives NuD≈0.023⋅5743⋅1.904≈250\mathrm{Nu}_D \approx 0.023 \cdot 5743 \cdot 1.904 \approx 250. Outside the stated ranges, for example in laminar flow or with large property variations, the fit can be far off, and other correlations such as that of Gnielinski are used.

Going deeper

For a student

From NuD\mathrm{Nu}_D one gets the heat transfer coefficient h=NuD k/Dh = \mathrm{Nu}_D \, k / D, where kk is the thermal conductivity. This coefficient then enters an energy balance of the tube or of a heat exchanger. The different exponents for heating and cooling account roughly for the change of viscosity with temperature near the wall.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Forced convection, concept, no evidence levelForced convectionconceptHeat transfer coefficient, concept, no evidence levelHeat transfer coeffic…conceptNusselt number, concept, no evidence levelNusselt numberconceptPrandtl number, concept, no evidence levelPrandtl numberconceptReynolds number, concept, no evidence levelReynolds numberconceptDittus–Boelter equation, statement, no evidence levelDittus–Boelter equati…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Forced convection concept
  2. Heat transfer coefficient concept
  3. Nusselt number concept
  4. Prandtl number concept
  5. Reynolds number concept
  6. Dittus–Boelter equation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Nusselt_number
About this page
Tag5X9Q3AABF3
URIm2://5X9Q3AABF3
Kindstatement, fact
WritingW0 stub
MSC80A19, 76R05, 76F10
StatusThis copy of the site has no build data. Run m2 status 5X9Q3AABF3.
AboutForced convection, Nusselt number
UsesNusselt number, Reynolds number, Prandtl number, Forced convection, Heat transfer coefficient
Changed3 November 2026

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