Prandtl Number Calculator

What is the Prandtl Number?

The Prandtl Number (Pr) is a dimensionless quantity in fluid dynamics that represents the ratio of momentum diffusivity to thermal diffusivity. Named after Ludwig Prandtl, it plays a crucial role in heat transfer and fluid flow analyses, providing insights into the relative thickness of the velocity and thermal boundary layers.

Formula

The Prandtl Number is calculated using the following equation:

\[ Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k} \]

Where:

• \( Pr \) = Prandtl Number (dimensionless)
• \( \nu \) = Kinematic viscosity (m²/s)
• \( \alpha \) = Thermal diffusivity (m²/s)
• \( c_p \) = Specific heat capacity at constant pressure (J/(kg·K))
• \( \mu \) = Dynamic viscosity (kg/(m·s))
• \( k \) = Thermal conductivity (W/(m·K))

Significance of Prandtl Number

The Prandtl Number provides valuable information about the relative importance of momentum and heat transfer in a fluid:

• Pr < 1: Thermal diffusivity dominates (e.g., liquid metals)
• Pr ≈ 1: Momentum and heat transfer are comparable (e.g., gases)
• Pr > 1: Momentum diffusivity dominates (e.g., oils)

Example Calculation

Let's calculate the Prandtl Number for water at 20°C:

1. Given:
   • Kinematic viscosity (\( \nu \)) = 1.004 × 10⁻⁶ m²/s
   • Thermal diffusivity (\( \alpha \)) = 1.43 × 10⁻⁷ m²/s
2. Apply the Prandtl Number formula: \[ Pr = \frac{\nu}{\alpha} \]
3. Substitute the known values: \[ Pr = \frac{1.004 \times 10^{-6} \text{ m²/s}}{1.43 \times 10^{-7} \text{ m²/s}} \]
4. Perform the calculation: \[ Pr \approx 7.02 \]

Visual Representation

The following diagram illustrates the concept of the Prandtl Number:

This diagram shows:

• A fluid volume (blue rectangle)
• Thermal diffusivity (\( \alpha \)) represented by the horizontal green dashed line
• Momentum diffusivity (kinematic viscosity, \( \nu \)) represented by the vertical red dashed line
• The Prandtl Number as the ratio of these two properties

Applications

The Prandtl Number is widely used in various fields, including:

• Heat exchanger design
• Boundary layer analysis
• Computational fluid dynamics (CFD)
• Natural convection studies
• Atmospheric science