Introduction & Context

In process engineering, the convective heat transfer coefficient h is a critical parameter for designing heat exchangers, reactors, and piping systems. Because h is not a fundamental property of a fluid, it must be determined through empirical correlations that relate the flow regime to the thermal boundary layer. Dimensionless groups—specifically the Reynolds (Re), Prandtl (Pr), and Nusselt (Nu) numbers—provide a generalized framework to calculate this coefficient across different scales and fluid types.

This calculation is typically used in the design phase of thermal systems to ensure that heat transfer rates meet process requirements while maintaining flow within desired regimes (e.g., ensuring turbulent flow for enhanced mixing and heat transfer). It is specifically applicable to steady, fully developed internal pipe flow. For the Dittus-Boelter correlation, fluid properties in the Reynolds and Prandtl numbers are evaluated at the bulk fluid temperature \(T_{b}\). The film temperature \(T_{f}\) is introduced for evaluating fluid properties used in the Grashof number calculation when assessing buoyancy effects.

Methodology & Formulas

The calculation follows a systematic approach to characterize the fluid flow and heat transfer regime. First, the film temperature is determined as the arithmetic mean of the bulk fluid temperature and the pipe wall temperature. This temperature is used for evaluating fluid properties in the Grashof number:

\[ T_{f} = \frac{T_{b} + T_{s}}{2} \]

The flow regime is characterized by the Reynolds number, which represents the ratio of inertial forces to viscous forces. All fluid properties in this expression are evaluated at the bulk temperature \(T_{b}\):

\[ Re = \frac{\rho \cdot V \cdot D}{\mu} \]

The Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity, characterizing the relationship between the velocity and thermal boundary layers. Fluid properties are evaluated at the bulk temperature \(T_{b}\):

\[ Pr = \frac{\mu \cdot C_{p}}{k} \]

For turbulent forced convection, the Nusselt number is calculated using the Dittus‑Boelter correlation for turbulent flow in pipes. This empirical relation is valid for heating or cooling scenarios, with the exponent n adjusted accordingly ( n = 0.4 for heating where \(T_{s} > T_{b}\), n = 0.3 for cooling where \(T_{s} < T_{b}\)).

\[ Nu = 0.023 \cdot Re^{0.8} \cdot Pr^{n} \]

Once the Nusselt number is determined, the convective heat transfer coefficient h is derived from the definition of the Nusselt number:

\[ h = \frac{Nu \cdot k}{D} \]

To assess the influence of natural convection (buoyancy) on the forced flow, the Grashof number is calculated. All fluid properties in the Grashof number are evaluated at the film temperature \(T_{f}\), and the temperature difference is defined as \(\Delta T = |T_{s} - T_{b}|\):

\[ Gr = \frac{g \cdot \beta \cdot \Delta T \cdot D^{3} \cdot \rho^{2}}{\mu^{2}} \]

For ideal gases, the volumetric thermal expansion coefficient is \(\beta = 1 / T_{f}\) where \(T_{f}\) is expressed in absolute units (Kelvin or Rankine). For liquids, \(\beta\) must be obtained from property tables at the film temperature.

Parameter	Condition / Threshold	Regime / Validity
Reynolds Number (Re)	Re ≥ 10,000	Turbulent Flow (Dittus-Boelter valid)
Prandtl Number (Pr)	0.6 ≤ Pr ≤ 160	Dittus-Boelter valid range
Length-to-Diameter (L/D)	L/D ≥ 10	Fully developed flow assumption
Buoyancy Ratio (Gr/Re²)	Gr/Re² < 0.1	Pure forced convection (negligible buoyancy)

Dimensionless groups allow process engineers to maintain geometric and dynamic similarity between laboratory-scale models and full-scale industrial equipment. By ensuring these ratios remain constant, you can predict performance without needing to replicate the exact physical dimensions of the final system. Key benefits include:

Reducing the number of variables required for experimental correlation.
Simplifying the analysis of complex fluid flow and thermal interactions.
Enabling the use of empirical correlations across different fluid properties and flow regimes.

While both involve heat transfer coefficients, they serve distinct purposes in thermal analysis:

Use the Nusselt number (Nu) to characterize the ratio of convective to conductive heat transfer at a fluid-solid boundary. It is primarily used to calculate the convective heat transfer coefficient (h).
Use the Biot number (Bi) to determine the validity of the lumped capacitance method. It compares the internal thermal resistance of a solid to the external convective resistance. If Bi is less than 0.1, you can assume a uniform temperature distribution within the solid.

The Prandtl number (Pr) represents the ratio of momentum diffusivity to thermal diffusivity. For process engineers, it provides critical insight into the relative thickness of the velocity boundary layer and the thermal boundary layer:

If Pr is high (e.g., viscous oils, Pr ≫ 1), the thermal boundary layer is much thinner than the velocity boundary layer.
If Pr is low (e.g., liquid metals, Pr ≪ 1), heat diffuses much faster than momentum, and the thermal boundary layer extends well beyond the velocity boundary layer.
If Pr ≈ 1 (e.g., most gases at ~0.7), the two boundary layers are of comparable thickness.
Understanding Pr is vital for selecting the correct correlation for the Nusselt number, as most correlations are highly sensitive to the fluid's Prandtl number.

The Grashof number (Gr) is used to quantify the ratio of buoyancy forces to viscous forces within a fluid. You must include this parameter when:

Natural convection is the dominant or a significant mode of heat transfer.
The flow velocity is low enough that forced convection correlations are no longer accurate.
You need to determine whether mixed convection effects are present by evaluating the buoyancy ratio Gr/Re². If this ratio exceeds 0.1, natural convection significantly assists or opposes the forced flow.
You are calculating the Rayleigh number (Ra = Gr × Pr), which is the standard criterion for determining the onset of turbulence in natural convection scenarios.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle

Dimensionless Groups in Heat Transfer

Introduction & Context

Methodology & Formulas

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