Nusselt Number (Nu): Convective Enhancement of Heat Transfer

Interpretation via boundary layers

Consider a wall at temperature T_w with free stream at T_∞. The local heat flux q″ = h (T_w − T_∞). In a conduction-only reference (no bulk motion), q″_cond ≈ k (T_w − T_∞) / L. Their ratio yields Nu, which can also be expressed as a dimensionless temperature gradient at the wall in similarity solutions. Thus Nu = 1 indicates pure conduction; Nu > 1 indicates convective enhancement.

Local vs average values

For external flows, a local Nu_x = h_x x / k varies with distance from the leading edge; an average Nū over a length L is typically reported for design. Internal flows use L = D_h and mean h over a heated length.

Coupling with Re and Pr

Heat transfer depends on momentum and thermal transport. Correlations therefore include Re and Pr:

• Prandtl number Pr = ν / α = μ c_p / k compares momentum to thermal diffusivity.
• In turbulent flows, Nu often scales like Re^m Prⁿ with exponents near m ≈ 0.8, n ≈ 0.3–0.4 for gases.

Internal laminar flow (fully developed)

For a circular tube:

• Constant wall temperature: Nu = 3.66.
• Constant heat flux: Nu = 4.36.
• If thermal entrance effects are significant, the Graetz number Gz = Re Pr D / L appears; entrance-length corrections (e.g., Sieder–Tate) modify Nu upward from the fully developed values.

Internal turbulent flow

Classic correlations include:

• Dittus–Boelter (smooth tubes, Re ≳ 10⁴, 0.7 ≲ Pr ≲ 160):

  Nu = 0.023 · Re^0.8 · Prⁿ, n ≈ 0.4 (heating), 0.3 (cooling).

• Gnielinski (wider validity, uses friction factor f):

  Nu = [(f / 8) · (Re − 1000) · Pr] / [1 + 12.7 · √(f / 8) · (Pr^2/3 − 1)].

  with f from the Moody relation and property evaluation at bulk/film conditions per guidance.

External flow over flat plates

For a laminar boundary layer (constant properties, negligible pressure gradient):

For a turbulent plate (transition at x_c):

with corrections for leading-edge laminar region. Surface roughness, pressure gradient, and property variation alter coefficients and exponents.

Natural (free) convection

Here Nu depends on Rayleigh Ra = g β (T_w − T_∞) L³ / (ν α) (or Grashof times Pr). Typical forms:

with n ≈ 1/4 (laminar) or 1/3 (turbulent) for canonical geometries (vertical plates, horizontal cylinders). Exact coefficients depend on orientation and boundary conditions.

Determining h

From an energy balance:

after correcting for axial conduction, radiation, contact resistances, and parasitic losses. Accurate Nu requires:

• Calibrated heat input and heat-loss accounting.
• Wall and fluid temperatures measured with minimal conduction errors (thin-film RTDs, embedded thermocouples, or IR diagnostics with known emissivity).
• Property evaluation at representative temperatures (often the film temperature (T_w + T_∞) / 2).

Data reduction choices

State whether local or average Nu is reported, the characteristic length L, and how properties were evaluated. Include uncertainties from instrumentation, property correlations, and geometric tolerances.

Heat exchangers and energy systems

Nu correlations drive sizing of shell-and-tube, plate, and micro-channel heat exchangers. Enhancements via fins, ribbing, or swirl-generators aim to increase Nu while balancing pressure-drop penalties (pumping power).

Electronics cooling and thermal management

From heat sinks with forced convection to natural-convection enclosures, designers select geometries to maximize Nu under tight volume and noise constraints. At small scales, entrance effects and property variations matter.

Process and chemical engineering

Reactors, evaporators, and condensers rely on accurate Nu prediction under multi-phase, fouling, and non-Newtonian conditions; correlations must be chosen and validated for the specific regime.

Built environment and HVAC

Room air distribution, radiant panels, and façade convection use Nu–Ra–Pr frameworks to predict comfort and energy use. Mixed convection (combined forced and natural) requires composite correlations or CFD with validated wall models.