Nusselt Number Calculator
Compute the Nusselt number Nu = h·L/k — the ratio of convective to conductive heat transfer — solving for any of its four variables with full unit flexibility. Includes live heated-plate schematic, application comparison, and full reference tables.
Nu compares convection to pure conduction
The Nusselt number answers a simple physical question: how much does fluid motion enhance heat transfer compared to what pure conduction alone would achieve across the same distance L? If the fluid is completely still, heat transfer is pure conduction and Nu = 1. If fluid motion is vigorous (as in a fan-cooled heat sink or a fast-flowing pipe), the convective enhancement can be 100×, 1,000×, or even 10,000× greater — and Nu takes those values. This is why the Nusselt number is a measure of convection strength: it does not tell you how heat moves, but how effectively the moving fluid moves it.
L must be chosen from the geometry, not the problem
Unlike h and k, which are properties of the fluid and flow, L is a chosen quantity that depends on the geometry of the problem. For a flat plate in parallel flow, L is the plate length in the flow direction. For a cylinder in cross-flow, L is the diameter. For a sphere, L is the diameter. For a duct or pipe, L is the hydraulic diameter Dh = 4A/P. For natural convection on a vertical plate, L is the height. The same physical situation with L chosen differently gives a different Nu — but the dimensionless correlations that relate Nu to Re and Pr are only valid for the specific L they were derived with. Using the wrong L is one of the most common mistakes in heat transfer calculations.
h and k are physically different — one is flow, one is fluid
The thermal conductivity k is a property of the fluid alone — you look it up in a table for air, water, or oil at the relevant temperature, and it does not change with flow speed. The convective coefficient h is the combined result of fluid properties (k, μ, cp, ρ) and flow conditions (velocity, turbulence, geometry) — you cannot look it up, you must either measure it or compute it from a correlation. This is exactly why Nu is useful: it packs all the fluid properties and flow conditions into one dimensionless number that can be correlated against Re and Pr, giving engineers a compact way to predict h for any geometry once the correlation has been established.
Every convection correlation is built on Nu
The entire discipline of convective heat transfer is organised around dimensionless correlations of the form Nu = f(Re, Pr). The Dittus-Boelter correlation for turbulent pipe flow is Nu = 0.023·Re0.8·Prn. The Churchill-Bernstein correlation for a cylinder in cross-flow is a more complex function of Re and Pr. Free convection correlations typically take the form Nu = C·(Gr·Pr)n where Gr is the Grashof number. Once you can compute Re and Pr (both measurable), you plug them into the correlation and out comes Nu, which immediately gives h = Nu·k/L. This is why the Nusselt number is the central dimensionless group of convective heat transfer.
How to Use This Calculator
The Nusselt number is a product of the convective coefficient and length, divided by thermal conductivity. Pick what to solve for, and enter the other three.
1. Choose what to solve for
Select Nu, h, L, or k from the toggle. All four rearrangements of Nu = h·L/k are algebraic and exact.
2. Use a preset for h
The h presets span the practical range: natural convection in air (10), forced air (50), water flow (1000), fast water (5000), boiling (15000), and condensing (50000) — all in W/(m²·K).
3. Pick the characteristic length
Choose L based on the geometry: plate length for a flat plate, diameter for a cylinder or sphere, hydraulic diameter for a duct. The presets span 1 cm to 100 m.
4. Match units carefully
Any consistent unit system works. SI is cleanest: h in W/(m²·K), L in m, k in W/(m·K) — the units cancel and Nu is dimensionless. The tool converts everything to SI internally, so unit switching does not change the answer.
5. Read the result and the Nu-scale
The result card shows Nu along with a colour-coded verdict classifying the flow regime — from pure conduction (Nu ≈ 1) through natural convection, laminar forced convection, turbulent forced convection, and on to boiling/condensing (Nu > 1000).
6. Switch to Advanced for depth
Advanced mode adds a k chart, a comparison showing typical Nu for ten applications (from a still-air plate to a boiling surface), a sensitivity panel, and companion metrics including convective heat flux, boundary layer thickness, and the ratio to conduction.
What This Tool Offers
Built around the Nusselt number definition Nu = h·L/k, the fundamental dimensionless group of convective heat transfer.
Four-way solving
Solve for Nu, h, L, or k. All four algebraic rearrangements are exact, and the tool computes convective heat flux and boundary layer thickness as bonus outputs.
Full unit flexibility
h in W/(m²·K), kW/(m²·K), BTU/(h·ft²·°F), or cal/(s·cm²·°C). L in m, cm, mm, ft, or in. k in W/(m·K), kW/(m·K), cal/(s·cm·°C), or BTU/(h·ft·°F).
Live heated-plate schematic
A flat plate with a growing thermal boundary layer, free-stream flow arrows, and a summary box showing the calculation — with the boundary layer thickness drawn to relative scale.
Presets and application comparison
One-click presets for h (six ranges), L (six scales), and k (six fluids). Advanced mode shows typical Nu across ten applications from still-air plates to boiling surfaces.
Three coordinated charts
Nu vs h (linear), Nu vs L (linear), and Nu vs k (hyperbolic) — each with your operating point marked and boundary-layer visual reference.
Reference tables
Typical Nu values by flow regime, heat transfer coefficient ranges for different convection modes, characteristic length selection guide, and a complete variable reference.
Reference Tables
Typical Nu values by regime, typical h values by convection mode, and the characteristic length selection guide.
Typical Nusselt Numbers by Flow Regime
Order-of-magnitude values for common convection situations. Actual values depend on geometry, Re, and Pr.
| Regime | Typical Nu | Dominant physics | Notes |
|---|
Typical Convective Heat Transfer Coefficients
Representative values of h for different convection modes. Values are ranges; the actual h depends on geometry and flow conditions.
| Convection mode | h W/(m²·K) | h BTU/(h·ft²·°F) | Typical examples |
|---|
Characteristic Length Selection Guide
Which L to use for common geometries. Choosing the right L matters — it determines which correlation applies.
| Geometry | Characteristic length L | Example correlation |
|---|---|---|
| Flat plate, parallel flow | Plate length in flow direction | ReL = ρ·U·L/μ, Nu = f(ReL, Pr) |
| Flat plate, perpendicular flow | Plate width | Use cylinder correlation with equivalent diameter |
| Cylinder in cross-flow | Diameter D | ReD = ρ·U·D/μ (Churchill-Bernstein) |
| Sphere in flow | Diameter D | ReD = ρ·U·D/μ (Whitaker) |
| Circular pipe or duct | Hydraulic diameter Dh = 4A/P | ReD = ρ·U·Dh/μ (Dittus-Boelter, Gnielinski) |
| Non-circular duct | Hydraulic diameter Dh = 4A/P | ReD = ρ·U·Dh/μ |
| Vertical plate, natural convection | Height H | RaH = g·β·ΔT·H³/(να), Nu = f(RaH, Pr) |
| Horizontal plate, natural convection (top) | Area / Perimeter ≈ L | RaL = g·β·ΔT·L³/(να) |
| Horizontal cylinder, natural convection | Diameter D | RaD = g·β·ΔT·D³/(να) |
| Sphere, natural convection | Diameter D | RaD = g·β·ΔT·D³/(να) |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| Nu | Nusselt number | 0.1 – 100,000 | dimensionless |
| h | Convective heat transfer coefficient | 1 – 100,000 | W/(m²·K) |
| L | Characteristic length | 0.001 – 100 | m |
| k | Thermal conductivity | 0.01 – 500 | W/(m·K) |
| Re | Reynolds number | 0.01 – 10⁸ | dimensionless |
| Pr | Prandtl number | 0.001 – 100,000 | dimensionless |
| Gr | Grashof number | 10³ – 10¹² | dimensionless |
| Ra | Rayleigh number = Gr·Pr | 10³ – 10¹² | dimensionless |
| q" | Convective heat flux | 1 – 10⁷ | W/m² |
What Does the Nusselt Number Measure?
The Nusselt number is the dimensionless ratio of convective to conductive heat transfer across a characteristic length L. It answers the question: how much does the motion of a fluid enhance heat transfer compared to what pure molecular conduction alone would achieve across the same distance? When Nu = 1, the fluid is effectively stationary and heat moves by pure conduction. When Nu is large, the fluid motion multiplies heat transfer many times over — through the combined effects of boundary-layer thinning, turbulence, and bulk transport of thermal energy. Every convection correlation ever published is ultimately a prediction of the Nusselt number for a specific geometry and flow regime.
Nusselt Number Formula
The defining relationship is a simple ratio:
where h is the convective heat transfer coefficient, L is the characteristic length, and k is the thermal conductivity of the fluid. Rearranged to solve for the other three variables:
L = Nu · k / h
k = h · L / Nu
Physically, Nu is the ratio of the actual heat transfer to the heat transfer that would occur by pure conduction across the same length:
The standard turbulent pipe flow correlation (Dittus-Boelter) illustrates the connection between Nu, Re, and Pr:
Natural convection correlations use the Rayleigh number instead:
Nusselt Number Variables
| Symbol | Meaning | Notes |
|---|---|---|
| Nu | Nusselt number | Dimensionless; ratio of convection to conduction |
| h | Convective coefficient | Depends on fluid AND flow |
| L | Characteristic length | Chosen from the geometry, not the fluid |
| k | Thermal conductivity | Property of the fluid alone |
| Re | Reynolds number | Ratio of inertial to viscous forces |
| Pr | Prandtl number | Ratio of momentum to thermal diffusivity |
| Ra | Rayleigh number | Gr·Pr; used in natural convection |
How Each Variable Affects the Nusselt Number
The equation is linear in h and L, and inverse in k. Doubling the convective coefficient doubles Nu; doubling the characteristic length doubles Nu; doubling the thermal conductivity halves Nu. This is why the Nusselt number is high for vigorous forced convection in fluids with low thermal conductivity, and low for gentle natural convection in fluids with high thermal conductivity. For example, water flowing at 1 m/s over a 1 m plate can give Nu ≈ 1000, while still air over the same plate gives Nu ≈ 10. The same geometry and same fluid property (k), but a 100× difference in Nu because the convective coefficient differs by 100×.
Because Nu is dimensionless, it is a property of the physical configuration rather than of any particular unit system. This makes it the natural quantity for experimental correlation and engineering prediction. Once a correlation Nu = f(Re, Pr) has been established for a geometry, it can be applied to any fluid and any scale, as long as Re and Pr match the range over which the correlation was validated. This is the essence of similarity analysis in heat transfer — the same dimensionless numbers at different scales produce the same physics.
Nusselt Number Examples
Nu = 1000 × 1 / 0.6 = 1666.7 (forced convection, strong enhancement)
Nu = 10 × 0.3 / 0.026 = 115.4 (natural convection, moderate enhancement)
Nu = 50 × 0.1 / 0.026 = 192.3
Nu = 15000 × 0.05 / 0.6 = 1250
h = Nu · k / L = 200 × 0.6 / 0.5 = 240 W/(m²·K)
L = Nu · k / h = 500 × 0.6 / 800 = 0.375 m
k = h · L / Nu = 500 × 0.2 / 100 = 1.0 W/(m·K)
Frequently Asked Questions
Both look like h·L/k, but they mean very different things. The Nusselt number uses the thermal conductivity of the fluid — it describes the ratio of convective to conductive transport within the fluid. The Biot number uses the thermal conductivity of the solid — it describes the ratio of internal conduction resistance in a solid object to external convection resistance at its surface. Same formula, different k, completely different physical interpretation. Nu tells you how strongly convection enhances heat transfer inside the fluid; Bi tells you whether the temperature inside a solid object can be treated as uniform (Bi < 0.1) or whether internal temperature gradients matter (Bi > 0.1). This is one of the most commonly confused pairs of dimensionless numbers in heat transfer.
Pure conduction through a layer of thickness L with conductivity k and temperature difference ΔT carries a heat flux of q" = k·ΔT/L. Convection across the same layer at the same ΔT carries q" = h·ΔT. If Nu = h·L/k = 1, then h = k/L, and the convective flux equals the conductive flux. That happens when the fluid is essentially stationary — no flow, no mixing, no boundary-layer thinning. In practice Nu = 1 is achieved only in perfectly still fluid, but it is the baseline against which all convection is measured. Every Nu > 1 represents heat transfer enhancement caused by fluid motion, and the magnitude of Nu tells you the multiplicative enhancement factor over pure conduction.
For air, natural convection over a vertical plate 1 m tall gives Nu ≈ 100–300 depending on ΔT. Forced air at 5–20 m/s over a similar plate gives Nu ≈ 500–2,000. For water, natural convection over a similar plate gives Nu ≈ 300–800. Forced water at 1–5 m/s through a pipe with diameter 0.05 m gives Nu ≈ 200–1,500 depending on Re. Boiling water on a hot surface can give Nu ≈ 1,000–5,000. Condensing steam on a vertical tube typically gives Nu ≈ 1,000–3,000. The general rule is that Nu grows with Re (stronger flow) and grows with Pr for high-Pr fluids (viscous oils), but the exact values always depend on geometry, flow regime, and the specific correlation used.
It depends entirely on the geometry and the correlation you plan to use. For a flat plate in parallel flow, L is the plate length in the direction of flow. For a cylinder in cross-flow, L is the diameter. For a sphere, L is the diameter. For a circular pipe, L is the inside diameter. For a non-circular duct, L is the hydraulic diameter Dh = 4A/P where A is the cross-sectional area and P is the wetted perimeter. For natural convection on a vertical plate, L is the height. Each correlation specifies its characteristic length explicitly — always read the correlation carefully and use the L it specifies, not some other plausible length. The same physical situation with L chosen incorrectly gives a Nu that does not correspond to any valid correlation.
For forced convection, the universal relationship is Nu = f(Re, Pr). The Reynolds number captures the flow regime — whether the flow is laminar, transitional, or turbulent — and how strongly the flow disturbs the thermal boundary layer. The Prandtl number captures the fluid's tendency to transport momentum versus heat. Together they determine the structure of the thermal boundary layer, which in turn determines Nu. The classical Dittus-Boelter correlation Nu = 0.023·Re0.8·Prn works for turbulent pipe flow with 0.7 ≤ Pr ≤ 160 and Re > 10,000. For natural convection, Re is replaced by the Grashof number (buoyancy-driven flow has no imposed velocity), and the combination Gr·Pr = Ra is the driving dimensionless group.
In principle, Nu < 1 would mean that fluid motion is reducing heat transfer below the pure-conduction limit, which is not physical for ordinary single-phase convection. Nu = 1 is the theoretical floor, and Nu > 1 in every real convective situation. In practice, some correlations produce Nu < 1 in extreme parameter regimes (very low Gr for natural convection, or very low Re for forced convection), but these are usually outside the correlation's stated validity range. When you see Nu < 1 from a correlation, it usually means the correlation is being applied outside its valid regime, or that conduction dominates to such an extent that the fluid can be treated as motionless. For engineering work, treat Nu = 1 as the natural floor.
Fluid properties — especially viscosity, thermal conductivity, and specific heat — vary with temperature. In a convection problem, the fluid temperature varies from Twall at the surface to T∞ in the free stream, so there is no single temperature at which to evaluate them. The standard convention is to use the film temperature Tfilm = (Twall + T∞)/2, which represents an average across the boundary layer. Correlations are then written with the understanding that all properties are evaluated at Tfilm. This convention works well when the temperature difference is moderate; for very large ΔT, more sophisticated correlations use property ratios like (μ∞/μwall) or (Pr∞/Prwall) to correct for the variation.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| Nu | Nusselt number | |
| h | Convective heat transfer coefficient | W/m2.K |
| L | Characteristic length | m |
| k | Fluid thermal conductivity | W/m.K |
What it means
The Nusselt number is a dimensionless parameter that represents the enhancement of heat transfer due to convection compared to conduction alone. It is defined as Nu = h L / k, where h is the convective heat transfer coefficient, L is a characteristic length (e.g., pipe diameter or plate length), and k is the thermal conductivity of the fluid. In forced convection, Nu is often correlated with Re and Pr via empirical equations (e.g., Dittus‑Boelter). A higher Nu indicates more effective convective heat transfer. This number is used in the design of heat exchangers, cooling of electronic components, and many thermal engineering applications. It allows engineers to scale up from experimental data to full‑scale equipment. The Nusselt number is named after Wilhelm Nusselt, who made significant contributions to heat transfer theory. It is a central concept in convective heat transfer analysis.
Worked example
Nusselt Number – Two Examples
Real‑World| Parameter | Value |
|---|---|
| h | 500 W/m²K |
| L | 0.05 m |
| k | 0.6 W/mK |
| Parameter | Value |
|---|---|
| h | 1000 |
| L | 0.1 |
| k | 0.6 |
Common mistakes
- Characteristic length L: Depends on geometry (e.g., D for a pipe, L for a flat plate). Choose the correct one.
- Heat transfer coefficient h: The average coefficient over the surface, not local.
- Thermal conductivity k: At the film temperature (average of surface and fluid temperatures).
- Units: h in W/(m²·K), L in m, k in W/(m·K) → Nu is dimensionless.
- Physical meaning: Nu = 1 means pure conduction; larger Nu indicates convection enhancement.
Applications
The Nusselt number, Nu = h·L/k, is the ratio of convective to conductive heat transfer across a fluid boundary layer. It is a key dimensionless parameter in convective heat transfer correlations. Engineers use Nu to calculate the convective heat transfer coefficient h from empirical correlations for different geometries (e.g., flow inside pipes, over flat plates, around spheres). Accurate h values are essential for sizing heat exchangers, cooling systems, and electronic devices. The Nusselt number depends on Reynolds and Prandtl numbers, and its relationship is determined experimentally or via numerical simulations. By applying Nu correlations, engineers can predict heat transfer rates, design effective cooling systems, and ensure thermal management in a wide range of applications.
- Calculation of convective heat transfer coefficients in pipes, ducts, and external flows
- Design of heat exchangers and cooling systems
- Thermal analysis of electronics and power electronics
- HVAC system design and comfort analysis
- Natural and forced convection studies in buildings and equipment
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