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What is a Nusselt Number? | Process Engineering Glossary
What is a Nusselt Number?
In piping engineering and process engineering, the Nusselt number (Nu) is a dimensionless parameter that quantifies the ratio of convective heat transfer to conductive heat transfer across a fluid boundary layer. Named after the German engineer Wilhelm Nusselt, it is defined as:
Nu = h × L / k
Where h is the convective heat transfer coefficient (W/m²·K), L is the characteristic length of the geometry (m), and k is the thermal conductivity of the fluid (W/m·K). Rearranging this definition gives the engineering application of the Nusselt number directly:
h = Nu × k / L
This is how the Nusselt number is used in practice: Nu is first calculated from an empirical correlation as a function of the Reynolds number, Prandtl number, and geometry. The film coefficient h — which is what the engineer actually needs to calculate pressure drop on the thermal side, size a heat exchanger, or verify a reboiler — is then extracted from the Nusselt number using the fluid thermal conductivity and the characteristic length.
The Nusselt number is the heat transfer analogue of the Sherwood number in mass transfer. A Nusselt number of 1 corresponds to pure conduction through a stagnant fluid layer with no convective enhancement. Values significantly above 1 indicate that convection is enhancing heat transfer beyond what the fluid’s thermal conductivity alone would provide. Turbulent pipe flow typically gives Nu of 50 to 500; laminar pipe flow in the fully developed regime gives Nu of 3.66 (constant wall temperature) or 4.36 (constant heat flux) — the exact theoretical result for Newtonian laminar flow.
Applications of the Nusselt Number
Fin-and-Tube Air Cooler Design
Air-cooled heat exchangers use empirical j_H = a × Re^b correlations for specific fin geometries (plain fins, wavy fins, louvred fins), where j_H = Nu / (Re × Pr^(1/3)). These correlations are obtained from wind tunnel testing of the specific fin array and are proprietary to each manufacturer. The Nu extracted from j_H then gives the air-side h, which is the dominant thermal resistance in an air cooler (air-side h of 30 to 120 W/m²·K versus tube-side h of 1,000 to 8,000 W/m²·K). Getting the air-side Nu correct from the manufacturer’s correlation is critical for air cooler sizing.
Process Reactor Jacket and Coil Heat Transfer
Jacketed reactor and tank heat transfer uses Nu correlations based on the agitator Reynolds number Re_a = ρ × N × D_I² / μ:
Nu = C × Re_a^(2/3) × Pr^(1/3) × (μ_bulk/μ_wall)^0.14 × (D_I/T)^a × (H/T)^b
Where N is the agitator speed, D_I is the impeller diameter, T is the tank diameter, and H is the liquid height. C, a, and b are geometry-dependent constants from experimental correlations for specific impeller types. This correlation gives the inside wall film coefficient h_inside for the process-side heat transfer in jacketed or coiled vessels.
Condensate Film Coefficient on Tube Exteriors
The Nusselt condensation model for laminar condensate flow on a vertical surface gives:
Nu = h_cond × L / k_L = 0.943 × [ρ_L × (ρ_L − ρ_V) × g × k_L³ × h_fg / (μ_L × L × ΔT)]^0.25
This analytical result — derived by Nusselt himself — remains the standard first-pass correlation for condensation film coefficients on vertical tubes and surfaces. The actual condensation coefficient in industrial condensers typically exceeds this Nusselt prediction by 10 to 100 percent because turbulent film flow, surface rippling, and droplet entrainment all enhance the actual heat transfer above the smooth laminar film assumption.
Benefits of the Nusselt Number Concept
Enables correct reboiler and evaporator sizing. The nucleate boiling heat transfer coefficient — whether from the Rohsenow pool boiling correlation or the Chen flow boiling correlation — directly determines the boiling-side film coefficient in the overall heat transfer coefficient equation. Getting this coefficient right ensures that the reboiler or evaporator achieves its design duty at the design utility temperature without excessive safety margin.
Establishes the safe operating envelope. Knowing the CHF for the specific fluid and geometry allows the engineer to set the maximum allowable heat flux with a defined safety margin. This margin is the primary protection against departure from nucleate boiling and the equipment damage that follows.
Guides surface selection and enhancement. Understanding that nucleation site density and surface wettability control the nucleate boiling coefficient allows informed selection between plain, enhanced, and porous surface tubes — balancing the improvement in h against the additional cost and the sensitivity of the enhanced surface to fouling or scale deposition.
Limitations to Consider
Correlations are geometry-specific. The Dittus-Boelter correlation applies to fully developed turbulent flow in smooth circular pipes. It does not apply to non-circular ducts (without hydraulic diameter correction), to the entrance region (first 10 to 60 diameters), to rough pipes, to curved passages (where Dean number effects increase h by 10 to 50 percent above straight tube values), or to pipe fittings. Engineers must select the Nu correlation appropriate to the specific geometry — applying a pipe flow correlation to the shell side of a shell-and-tube exchanger, or a smooth tube correlation to a tube with internal fins, introduces serious error.
Properties must be evaluated at the correct temperature. The Nusselt number and its associated Prandtl number and Reynolds number must all be evaluated at a consistent reference temperature — usually the mean bulk temperature of the fluid in the passage, with a separate viscosity at the wall temperature for the Sieder-Tate correction. For fluids with strong temperature dependence of k, Cp, and μ (viscous oils, high-viscosity polymers), using properties at the wrong temperature can change Nu by 20 to 100 percent.
Transition flow requires special treatment. Between Re = 2,300 and Re = 10,000, the flow is neither fully laminar nor fully turbulent and standard correlations in either limit are unreliable. The Gnielinski correlation extends the range of applicability:
Nu = (f/8) × (Re − 1,000) × Pr / [1 + 12.7 × (f/8)^0.5 × (Pr^(2/3) − 1)] for 2,300 < Re < 5×10⁶
This correlation gives smooth transitions between the laminar and turbulent regimes and is preferred for preliminary designs where the expected Re may span the transition range.
Nusselt Number FAQ
What is the Nusselt number and how is it used to calculate the film coefficient in process engineering?
The Nusselt number Nu = h × L / k is the dimensionless ratio of convective to conductive heat transfer across a fluid boundary layer. Process engineering uses it to calculate the convective film coefficient h = Nu × k / L by first computing Nu from empirical correlations as a function of the Reynolds and Prandtl numbers, then extracting h from the fluid thermal conductivity and the characteristic length. Nu = 3.66 for fully developed laminar flow in a circular pipe at constant wall temperature — the exact theoretical minimum for any pipe flow configuration. Nu = 50 to 500 is typical for turbulent flow — the high values reflecting the intense fluid mixing that drives thermal energy from the bulk to the wall far more effectively than molecular conduction alone.
How does the Nusselt number enter heat exchanger design and the overall heat transfer coefficient?
For any heat exchanger, Nu is calculated for each fluid stream using the appropriate correlation (Dittus-Boelter for turbulent tube-side flow, Bell-Delaware j_H factor for shell-side crossflow, manufacturer’s correlation for plate heat exchanger channels). Each Nu gives a film coefficient h through h = Nu × k / L. These film coefficients, combined with the tube wall conduction resistance and fouling factors, give the overall heat transfer coefficient (U) from 1/U = 1/h_shell + R_wall + R_fouling + 1/h_tube. The side with the lowest h dominates U, which directs design effort — increasing gas-side velocity to raise Nu on the gas side of an air cooler has far more impact than improving the already-high liquid-side h.
How does the Nusselt number depend on Reynolds number, Prandtl number, and kinematic viscosity?
For turbulent pipe flow, the Dittus-Boelter correlation Nu = 0.023 × Re^0.8 × Pr^0.4 shows that Nu increases as the 0.8 power of Re and the 0.4 power of Pr. The Reynolds number Re = VD/ν depends on kinematic viscosity ν — higher ν (more viscous fluid at the same velocity and diameter) gives lower Re and lower Nu, reducing the film coefficient. Convection is weaker for more viscous fluids at the same velocity because the boundary layer is thicker (higher ν reduces Re, which thickens the boundary layer). The Prandtl number Pr = ν/α captures the ratio of kinematic viscosity to thermal diffusivity — higher Pr fluids (viscous oils) have thinner thermal boundary layers relative to their velocity boundary layers, which partially offsets the low Re penalty but does not compensate fully, which is why viscous fluids consistently give lower h at the same mean velocity than water or other low-viscosity fluids at the same conditions.
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