Prandtl Number

Overview

Engineering reference data for Prandtl Number in heat transfer.

Key Formulas

Fourier's Law

$q = -k \nabla T$

Heat flux proportional to temperature gradient.

Convective Heat Transfer

$Q = hA(T_s - T_\infty)$

Heat transfer between surface and fluid.

Stefan-Boltzmann Law

$q = \varepsilon \sigma T^4$

Radiative heat flux from a surface.

Thermal Resistance

$R_{th} = \frac{L}{kA}$

Resistance to heat conduction.

Variables

Practical Example

The following calculation demonstrates how to compute the Prandtl number for a fluid with given properties.

Given:

• Dynamic viscosity, μ = 15 cP
• Specific heat capacity, cₚ = 0.50 Btu/lb·°F
• Thermal conductivity, k = 0.06 Btu/(h·ft²·°F/ft)

Step 1: Convert Dynamic Viscosity to Compatible Units

$$\mu = 15 \, \text{cP} \times \frac{2.4191 \, \text{lbm/(ft·hr)}}{1 \, \text{cP}} = 36.3 \, \text{lbm/(ft·hr)}$$

Step 2: Calculate the Prandtl Number Using the formula $Pr = \frac{\mu \cdot c_p}{k}$:

Pr = \frac{36.3 \, \frac{\text{lbm}}{\text{ft·hr}} \times 0.50 \, \frac{\text{Btu}}{\text{lbm·°F}}}{0.06 \, \frac{\text{Btu}}{\text{h·ft^{2}·°F/ft}}} = 302

Interpretation: A Prandtl number of 302 indicates a fluid (likely an oil) where momentum diffusivity dominates significantly over thermal diffusivity. Heat diffusion is slow relative to momentum transfer.

Typical Prandtl Number Ranges

The Prandtl number varies widely depending on the fluid type and its state.

Key Physical Insight

The Prandtl number ($Pr$) is a fluid property that compares the rate of momentum diffusion (viscous effects) to the rate of thermal diffusion (heat conduction). It is crucial for predicting heat transfer in convection.

• Pr ≪ 1 (e.g., liquid metals): Heat diffuses much faster than momentum. The thermal boundary layer is much thicker than the velocity boundary layer.
• Pr ≈ 1 (e.g., gases): Momentum and heat diffuse at similar rates. The boundary layers for velocity and temperature are nearly the same.
• Pr ≫ 1 (e.g., oils): Momentum diffuses much faster than heat. The velocity boundary layer is much thicker than the thermal boundary layer.

References

• Original Source