The Nusselt number calculator above handles the dimensionless group that sits at the centre of every convection problem. The Nusselt number, written Nu, is the ratio of convective heat transfer at a surface to the heat that would cross the same layer of the same fluid by conduction alone. It is defined as Nu = hL ÷ k, where h is the convective coefficient, L is the characteristic length of the geometry and k is the thermal conductivity of the fluid. A Nusselt number of 40 means the moving fluid is carrying heat away forty times faster than a stagnant layer of the same fluid would.
Arb Digital publishes free engineering calculators that state their assumptions plainly and cross-link the neighbouring quantity rather than pretending to be the only page you need. This one runs the definition in both directions and adds the Dittus–Boelter correlation, which is the most widely taught way to predict a Nusselt number for turbulent flow in a tube before any hardware exists to measure.
What This Nusselt Number Calculator Does
The definition Nu = hL/k can be read three ways, and the mode selector decides which. If you have a measured or specified film coefficient, the calculator returns the Nusselt number and tells you how much better than conduction your surface is doing. If you have a Nusselt number from a correlation or a textbook chart, it inverts to give the coefficient h = Nu k ÷ L, which is the number you actually put into a heat transfer calculation. If you have neither, the correlation route predicts Nu from Reynolds and Prandtl numbers.
Alongside the headline figure the grid shows the convective coefficient, the coefficient a pure conduction layer of the same thickness would have given, the thermal boundary layer thickness implied by the answer, and the film resistance per square metre. That last number is the one that goes into a series thermal resistance chain, and it is often more useful than h itself.
This page does not compute Reynolds or Prandtl numbers for you. The Reynolds number calculator takes velocity, length and viscosity and returns Re, and the Prandtl number calculator builds Pr from viscosity, specific heat and conductivity. Bring their outputs here.
How to Use It
- Choose the characteristic length the correlation intends. Dittus–Boelter is written for internal pipe flow and expects the hydraulic diameter. External flow over a plate expects distance from the leading edge. A correlation used with the wrong length is not slightly wrong, it is meaningless.
- Use fluid properties, not wall properties. The k in the Nusselt number is the fluid's conductivity. Putting a copper conductivity in because the pipe is copper produces a Nusselt number several hundred times too small.
- Evaluate properties at the film temperature. For most correlations that is the mean of the bulk fluid and wall temperatures. Water's conductivity rises about 15 % between 20 °C and 90 °C, and its viscosity falls by a factor of three, so the choice matters.
- Pick the right Prandtl exponent. Use 0.4 when the fluid is being heated and 0.3 when it is being cooled. The asymmetry exists because near-wall viscosity moves in opposite directions in the two cases and reshapes the velocity profile.
- Check the validity flags in the result note. The calculator tells you when the Reynolds or Prandtl number you entered sits outside the range Dittus–Boelter was fitted for.
The Formula and a Worked Example
The definition is Nu = hL ÷ k. Every term is in SI: h in watts per square metre per kelvin, L in metres, k in watts per metre per kelvin. The units cancel completely, which is what makes Nu dimensionless and therefore transferable between a laboratory rig and a full-size plant.
Take the defaults. A film coefficient of 250 W/m²·K on a 0.05 m characteristic length in water with k = 0.6 W/m·K gives Nu = 250 × 0.05 ÷ 0.6 = 20.83. The conduction-only coefficient for that same 0.05 m layer is k/L = 0.6 ÷ 0.05 = 12 W/m²·K, and 250 ÷ 12 is indeed 20.83. The film resistance is 1/h = 0.004 m²·K/W, and the thermal boundary layer thickness implied by h = k/δ is δ = 0.6 ÷ 250 = 2.4 mm.
The Dittus–Boelter correlation is Nu = 0.023 Re0.8 Prn. At Re = 50,000 and Pr = 5 with n = 0.4: Re0.8 = 5,744 and Pr0.4 = 1.9036, so Nu = 0.023 × 5,744 × 1.9036 = 251.5. Converting back through the definition with k = 0.6 and a 0.05 m tube gives h = 251.5 × 0.6 ÷ 0.05 = 3,018 W/m²·K, which is the right order for turbulent water in a tube of that size.
Note the exponent on Reynolds. Because Nu scales as Re0.8, doubling the flow rate raises the heat transfer coefficient by only 20.8 = 1.74 times while raising the pumping pressure drop by roughly four times. That asymmetry is why brute-force pumping is such an expensive way to improve a heat exchanger, and why surface enhancement usually wins instead.
How This Differs From the Other Dimensionless Groups
Four dimensionless numbers on this site look superficially alike and answer completely different questions, so it is worth stating each boundary in a line.
The Nusselt number compares convection at a surface to conduction through the fluid. Its conductivity term is the fluid's. The Biot number calculator compares convection at a surface to conduction inside the solid, and its conductivity term is the solid's. They share an algebraic form and nothing else: Biot answers "is my object internally uniform in temperature?" while Nusselt answers "how good is my film?". Get the two conductivities the wrong way round and both answers become nonsense.
The Prandtl number calculator gives a pure fluid property, the ratio of momentum diffusivity to thermal diffusivity, with no geometry and no flow in it at all. It is an input to Nusselt correlations, never an output. The Reynolds number calculator characterises the flow regime and is the other input. Nusselt is the output the two of them feed.
Finally, the heat transfer calculator takes a coefficient and an area and a temperature difference and returns watts. It is downstream of everything here. If you want the coefficient itself as the deliverable rather than the dimensionless group, the heat transfer coefficient calculator is the page built around h, and this page is the one built around the ratio that predicts it.
Where the Dittus–Boelter Correlation Is Valid
The correlation is an empirical fit, and it carries a published range of applicability that gets ignored far too often. It is intended for fully developed turbulent flow in smooth circular tubes with Reynolds numbers above roughly 10,000, Prandtl numbers between about 0.6 and 160, and a length-to-diameter ratio above about 10. It also assumes a moderate temperature difference between wall and bulk fluid, because it makes no viscosity-ratio correction.
Take it outside that envelope and the errors are large. Below Re = 10,000 the flow is transitional or laminar and the correlation over-predicts badly; laminar internal flow with a constant wall temperature converges on a constant Nu of about 3.66 regardless of Reynolds number, which no power law will reproduce. Liquid metals have Prandtl numbers around 0.01 and need their own correlations entirely, because the thermal boundary layer is far thicker than the velocity boundary layer. Viscous oils with large wall-to-bulk temperature differences need a viscosity-ratio correction such as the Sieder–Tate form.
Even inside its range, treat Dittus–Boelter as accurate to roughly plus or minus 25 %. It is a correlation fitted to scattered data, not a derivation. Entrance effects, surface roughness, tube curvature and non-uniform heating all move the real answer. MIT OpenCourseWare's Intermediate Heat and Mass Transfer course develops the boundary-layer analysis these correlations approximate, including where the power-law form comes from and why the exponents take the values they do.
Reading the Nusselt Number as Boundary Layer Thickness
There is a physical picture buried in the number that makes it far easier to reason about. If you define an effective thermal boundary layer thickness δ such that conduction across it gives the same heat flux as the real convection, then h = k/δ, and substituting into the definition gives Nu = L/δ. The Nusselt number is simply the characteristic length divided by the effective film thickness.
That reframing explains several things at once. A high Nusselt number means a thin film, which is exactly what turbulence produces by scrubbing the near-wall region. It explains why Nu grows with Reynolds number: faster flow thins the layer. It explains why the number is geometry-dependent: doubling the tube diameter doubles L and roughly doubles Nu at the same film thickness, even though the heat transfer coefficient has not improved at all. And it warns you that a Nusselt number quoted without its characteristic length is close to useless, which is why every published correlation names the length it uses in its first line.
Free and Forced Convection Are Different Problems
Everything above concerns forced convection, where a pump or fan sets the velocity and Reynolds number describes it. In free or natural convection the fluid moves only because heating changes its density, so there is no imposed velocity to build a Reynolds number from. Those correlations use the Rayleigh or Grashof number in place of Reynolds, typically in the form Nu = C Ram with exponents near a quarter for laminar and a third for turbulent conditions.
The definition of the Nusselt number itself is unchanged, so the first two modes on this page apply directly to natural convection results. Only the third mode, the Dittus–Boelter correlation, is restricted to forced turbulent internal flow. If your problem is a hot vertical wall in still air, take Nu from a natural convection correlation and use the "h from Nu" mode here to convert it into a coefficient. The thermal conductivity values that go into either route are tabulated in standards work such as the reference data published by NIST's fundamental physical constants and reference data programme, and the thermal conductivity calculator handles conduction through solid layers once you have the film coefficients at each face.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the wall's thermal conductivity instead of the fluid's — the Nusselt number's k is always the fluid property, and a metal value makes the result meaningless.
- Mismatching the characteristic length to the correlation — hydraulic diameter for internal flow, leading-edge distance for a plate, sphere diameter for a sphere. The correlation names its length for a reason.
- Running Dittus–Boelter below Re = 10,000 — in laminar internal flow the Nusselt number is roughly constant near 3.66 or 4.36 depending on boundary condition, and no power law reproduces that.
- Evaluating fluid properties at the bulk temperature only — most correlations specify the film temperature, and for viscous fluids the difference is not small.
- Confusing Nusselt with Biot — identical algebra, opposite conductivity. One tells you about the fluid film, the other about temperature uniformity inside the solid.
Related Free Tools From Arb Digital
Build the inputs with the Reynolds number calculator and the Prandtl number calculator, then bring the result here. Convert the answer into a usable coefficient with the heat transfer coefficient calculator, check whether a lumped model is even allowed with the Biot number calculator, and turn coefficients into watts with the heat transfer calculator. For transient problems the Newton's law of cooling calculator solves the exponential, the specific heat calculator supplies the energy storage term, and the thermal conductivity calculator handles the solid layers. Everything is listed on the free online tools hub.
Frequently Asked Questions
It is the ratio of heat actually convected away from a surface to the heat that would cross the same layer of the same fluid by conduction alone. Equivalently it is the characteristic length divided by the effective thermal boundary layer thickness. A value of 30 means the moving fluid removes heat thirty times faster than a stagnant film of that fluid would.
They share the same algebraic form but use different conductivities. The Nusselt number divides by the fluid's thermal conductivity and describes how good the convective film is. The Biot number divides by the solid's thermal conductivity and describes whether the solid is internally uniform in temperature. Swapping the two conductivities makes both answers wrong.
It can, and it usually signals a problem. Values below one mean convection is transferring less heat than conduction across the same length, which happens in genuinely stagnant fluid or in the low-Rayleigh limit of natural convection. Far more often it means the characteristic length or the conductivity was entered wrongly.
It is fitted for fully developed turbulent flow in smooth circular tubes, with Reynolds numbers above about 10,000, Prandtl numbers between roughly 0.6 and 160, a length-to-diameter ratio above about 10, and a moderate wall-to-bulk temperature difference. Even inside that envelope, expect accuracy of around plus or minus 25 per cent.
Because near-wall viscosity shifts in opposite directions in the two cases. Heating a liquid thins it near the wall and steepens the velocity profile there; cooling thickens it and does the reverse. The correlation absorbs that asymmetry with an exponent of 0.4 for heating and 0.3 for cooling rather than modelling the viscosity change directly.
Rearrange the definition to h equals Nu multiplied by the fluid's thermal conductivity, divided by the characteristic length. The second mode on this page does exactly that. Make sure the characteristic length you use is the same one the correlation that produced your Nusselt number was written for.
The definition modes do, because the definition of the Nusselt number does not care what drives the flow. The Dittus-Boelter mode does not, because it is a forced-convection correlation built on Reynolds number. For natural convection, take the Nusselt number from a Rayleigh or Grashof correlation and use the second mode here to convert it to a coefficient.
This tool is provided for educational and preliminary engineering use. Correlations are empirical fits with published ranges of validity and typical uncertainties of tens of per cent. Verify thermal designs against measured data, manufacturer performance curves or a validated simulation before committing to them.
