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Nusselt Number Calculator

Nusselt number equals convection coefficient times length divided by conductivity

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Nusselt Number Equation

The Nusselt number compares convective heat transfer at a surface to pure conduction through the fluid. A Nu of 1 means convection adds nothing beyond what conduction alone provides. Engineers use Nusselt correlations to determine the convection coefficient h without costly experiments.

Nu = hL / k

How It Works

The Nusselt number compares convective heat transfer at a surface to pure conduction through the fluid. A Nu of 1 means convection adds nothing beyond what conduction alone would provide. In practice, turbulent flows over heated surfaces produce Nu values of 100 or more, meaning convection is far more effective than conduction. Engineers use Nusselt correlations (functions of Reynolds and Prandtl numbers) to determine the convection coefficient h without costly experiments, enabling quick sizing of heat exchangers, cooling fins, and pipe insulation.

Example Problem

Water (k = 0.6 W/(m·K)) flows through a 0.02 m diameter pipe with a convection coefficient h = 3,000 W/(m²·K). What is the Nusselt number?

  1. Identify the known values: convection coefficient h = 3,000 W/(m²·K), characteristic length L = 0.02 m, thermal conductivity k = 0.6 W/(m·K).
  2. Determine what we are solving for: the Nusselt number Nu, which quantifies convective vs. conductive heat transfer.
  3. Write the Nusselt number equation: Nu = h × L / k.
  4. Substitute the known values: Nu = 3,000 × 0.02 / 0.6.
  5. Compute the numerator: 3,000 × 0.02 = 60.
  6. Divide by the thermal conductivity: Nu = 60 / 0.6 = 100. Convection transfers heat 100 times faster than conduction alone.

A Nusselt number of 100 means convection transfers heat 100 times faster than conduction alone across the same fluid layer, typical for turbulent water flow in a pipe.

When to Use Each Variable

  • Solve for Nusselt Number — when you know the convection coefficient, characteristic length, and thermal conductivity.
  • Solve for Convection Coefficient — when you know the Nusselt number from a correlation and need to find h for heat exchanger sizing.
  • Solve for Characteristic Length — when you know h, k, and Nu and need to determine the geometric scale.
  • Solve for Thermal Conductivity — when you know Nu, h, and L and need to back-calculate the fluid's thermal conductivity.

Key Concepts

The Nusselt number is the ratio of convective to conductive heat transfer across a boundary. Nu = 1 means no convective enhancement; turbulent flows routinely reach Nu > 100. Engineers use empirical correlations (Dittus-Boelter, Sieder-Tate, Churchill-Bernstein) that express Nu as a function of Reynolds and Prandtl numbers, allowing them to determine the convection coefficient without direct measurement.

Applications

  • Heat exchanger design: determining the convection coefficient to size tube bundles and shell passes
  • Electronics cooling: estimating heat dissipation from circuit boards and heat sinks under forced airflow
  • Chemical process engineering: sizing jacketed reactors and condensers for required thermal duty
  • HVAC: calculating heat transfer in ductwork and piping for building climate systems

Common Mistakes

  • Using the wrong characteristic length — pipe diameter for internal flow, plate length for external flow; the wrong choice invalidates the correlation
  • Applying a turbulent correlation to laminar flow — Dittus-Boelter requires Re > 10,000; for laminar pipe flow, Nu is a fixed constant (3.66 or 4.36)
  • Forgetting to evaluate fluid properties at the correct temperature — most correlations use bulk mean temperature, but Sieder-Tate uses a wall-to-bulk viscosity ratio

Frequently Asked Questions

What does a high Nusselt number mean for heat transfer?

A high Nusselt number means convection is transferring heat much faster than conduction alone. For example, Nu = 200 means the convective heat transfer rate is 200 times what pure conduction through the same fluid layer would deliver. Turbulent flows in industrial heat exchangers routinely produce Nu values of 100 to 500.

How is the Nusselt number related to the heat transfer coefficient?

The Nusselt number directly determines the convection heat transfer coefficient: h = Nu × k / L. If you know Nu from a correlation (like Dittus-Boelter) and the fluid’s thermal conductivity k, you can calculate h without any experiments. This is the primary engineering use of the Nusselt number.

What is the Nusselt number used for in heat exchanger design?

Engineers use Nusselt correlations to calculate the convection coefficient h without experiments. From h, they determine the overall heat transfer coefficient U and then size the heat exchanger area needed to achieve the required duty. Common correlations include Dittus-Boelter for turbulent pipe flow and Sieder-Tate for viscous fluids.

What is the Nusselt number for laminar flow in a pipe?

For fully developed laminar flow with constant wall heat flux, Nu = 4.36. With constant wall temperature, Nu = 3.66. These are exact analytical values and do not depend on Reynolds or Prandtl numbers, unlike turbulent flow correlations.

How does the Nusselt number relate to the Sherwood number?

The Sherwood number is the mass-transfer analogue of the Nusselt number. Nu describes convective heat transfer relative to conduction, while Sh describes convective mass transfer relative to diffusion. In systems where heat and mass transfer analogies apply, their correlations have the same functional form.

Why does the Nusselt number increase with turbulence?

Turbulent eddies mix hot and cold fluid far more efficiently than molecular conduction. This enhanced mixing dramatically increases the effective heat transfer coefficient. In turbulent pipe flow, Nu scales roughly as Re⁰⋅⁸ Pr⁰⋅⁴, so doubling the flow velocity can increase Nu by about 70%.

What is the Dittus-Boelter equation for the Nusselt number?

The Dittus-Boelter equation is Nu = 0.023 Re⁰⋅⁸ Prⁿ, where n = 0.4 for heating and n = 0.3 for cooling. It applies to fully developed turbulent flow in smooth circular tubes with Re > 10,000 and 0.6 < Pr < 160. It is the most widely used Nusselt correlation in engineering practice.

Nusselt Number Formula

The Nusselt number is a dimensionless ratio that compares convective heat transfer at a surface to pure conduction through the fluid:

Nu = h × L / k

Where:

  • Nu — Nusselt number (dimensionless)
  • h — convection heat transfer coefficient, in W/(m²·K)
  • L — characteristic length, in meters (m)
  • k — thermal conductivity of the fluid, in W/(m·K)

For internal pipe flow, L is the pipe diameter. For external flow over a flat plate, L is the plate length. The choice of characteristic length must match the correlation used (Dittus-Boelter, Churchill-Bernstein, etc.).

Worked Examples

Heat Exchanger Design

What is the tube-side Nusselt number for water in a shell-and-tube heat exchanger?

Water flows through a 25 mm diameter tube with h = 4,000 W/(m²·K). The thermal conductivity of water at 60 °C is k = 0.65 W/(m·K).

  • L = 0.025 m (tube diameter)
  • Nu = h × L / k = 4,000 × 0.025 / 0.65
  • Nu = 100 / 0.65
  • Nu = 153.85

This high Nusselt number indicates strong turbulent convection, typical for water at moderate velocities in heat exchanger tubes.

Solar Collector

What convection coefficient does a flat plate solar absorber achieve?

A solar collector has Nu = 35 (from a Nusselt correlation for forced convection over a flat plate), plate length L = 1.5 m, and uses air with k = 0.026 W/(m·K).

  • h = Nu × k / L = 35 × 0.026 / 1.5
  • h = 0.91 / 1.5
  • h = 0.607 W/(m²·K)

The low h value is typical for air — gases have much lower convection coefficients than liquids, which is why solar collectors often use selective coatings to reduce radiative losses.

Nuclear Reactor

What Nusselt number describes the coolant channel in a pressurized water reactor?

Pressurized water (k = 0.55 W/(m·K)) flows through a 12 mm diameter fuel assembly channel with h = 25,000 W/(m²·K).

  • L = 0.012 m (channel hydraulic diameter)
  • Nu = h × L / k = 25,000 × 0.012 / 0.55
  • Nu = 300 / 0.55
  • Nu = 545.45

Nu values above 500 are common in nuclear reactor coolant channels due to the extremely high flow velocities and temperatures required for safe heat removal.

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