Prandtl Number Calculator – Significance and Calculation

What is Prandtl number?

The Prandtl number (\( \text{Pr} \)) is a dimensionless number named after the German physicist Ludwig Prandtl. It is defined as the ratio of momentum diffusivity to thermal diffusivity. This number gives the relationship between kinematic viscosity and thermal diffusivity of the fluid.

Mathematically, the Prandtl number is given by:

\[\text{Pr} = \frac {\text{Momentum Diffusivity}} {\text{Thermal Diffusivity}}\]

\[\text{Pr} = \frac{\nu}{\alpha} = \frac{\mu \, C_p}{k}\]

where:

• \(\nu\) is the momentum diffusivity (kinematic viscosity) (m²/s)
• \(\alpha\) is the thermal diffusivity (m²/s)
• \(\mu\) is the dynamic viscosity (N·s/m²)
• \(C_p\) is the specific heat (J/kg·K)
• \(k\) is the thermal conductivity (W/m·K)

Note: The mass transfer analog of the Prandtl number (Pr) is the Schmidt Number.

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Prandtl Number Calculator

This calculator is helpful to calculate parameters like \(Pr\) Number, dynamic viscosity (μ), thermal conductivity (k), or specific heat capacity \(C_P\) based on the formula \(\text{Pr} = \frac{\mu \, C_p}{k}\) This tool is useful for engineers and scientists working in fields involving heat transfer and fluid dynamics.

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Prandtl Number Significance

The Prandtl number \(Pr\) is significant in Heat Transfer and Fluid Mechanics. It provides a measure of the relative rates of momentum and heat transport by diffusion in a fluid.

– Fluid Behavior Characterization

• The \(Pr\) number indicates the relative thickness of the velocity boundary layer to the thermal boundary layer.
• Low Prandtl number (Pr≪1): Thermal diffusivity dominates, meaning heat diffuses much faster than momentum (for liquid metals).
• High Prandtl number (Pr≫1): Momentum diffusivity dominates, meaning momentum diffuses much faster than heat (for oils)

– Boundary Layer Characteristics

• In fluids, Low \(Pr\) means, the thermal boundary layer is much thicker than the velocity boundary layer.
• In fluids, High \(Pr\) means the velocity boundary layer is thicker than the thermal boundary layer.

– Heat Transfer Analysis

• In convective heat transfer, \(Pr\) number helps to predict the pattern and efficiency of heat transfer.
• Along with Reynolds Number, it also calculates the Nusselt Number which characterizes the convective heat transfer coefficient.

Typical Ranges for Prandtl Number for various Fluids

Air at room temperature has a \(Pr = 0.71 \) and for water at 18°C it is around 7.56, which means that the thermal diffusivity is more dominant for air than for water.

For many fluids, \(\text{Pr}\) lies in the range from 1 to 10. For gases, \(\text{Pr}\) is generally about 0.7. For liquid metals the Prandtl number is very small, generally in the range from 0.01 to 0.001.

Source: Wikipedia – Prandtl Number – Experimental Values at specific temperature-Pressure conditions provided in the table below:

Substance	Temperature/Condition	Prandtl_Number (Pr)
Molten potassium	975 K	0.003
Mercury	Room temperature	~0.015
Molten lithium	975 K	0.065
Noble gas mixtures	Various	0.16–0.7
Oxygen	Room temperature	0.63
Air and many other gases	Room temperature	~0.71
Gaseous ammonia	Room temperature	1.38
R-12 refrigerant	Various	4–5
Water	18 °C	7.56
Seawater	0 °C	13.4
Seawater	20 °C	7.2
n-Butanol	Room temperature	50
Engine oil	Various	100–40,000
Glycerol	Room temperature	1000
Polymer melts	Various	10,000
Earth’s mantle	Various	~1×10²⁵

Example Problem

Calculate the Prandtl number of ethyl alcohol at 300K. Refer to Perry’s for the following values: dynamic viscosity (μ) in Pa·s, thermal conductivity (k) in W/(m·K), and specific heat capacity (Cp) in J/(kg·K)?

To solve the given problem, we need to calculate the \( \text{Pr} \) for ethyl alcohol at 300K using the provided properties:

Given:

• Dynamic viscosity (\( \mu \)) = 1.2 Pa·s
• Thermal conductivity (\( k \)) = 0.16 W/(m·K)
• Specific heat capacity (\( C_p \)) = 2440 J/(kg·K)

Using formula, \( \text{Pr} = \frac{\mu \cdot C_p}{k} \)

Substituting values into the formula,

\( \text{Pr} = \frac{1.2 \, \text{Pa·s} \times 2440 \, \text{J/(kg·K)}}{0.16 \, \text{W/(m·K)}} \)

\( \text{Pr} = \frac{1.2 \times 2440}{0.16} \)

\( \text{Pr} = \frac{2928}{0.16} \)

\( \text{Pr} = 18,300 \)

Therefore, the \(Pr\) number of ethyl alcohol at 300K is 18,300.

Resources

• “Perry’s Chemical Engineers’ Handbook” by Don W. Green and Marylee Z. Southard
• “Fundamentals of Heat and Mass Transfer” by Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, and Adrienne S. Lavine
• “Fluid Mechanics” by Frank M. White

Disclaimer: The Solver provided here is for educational purposes. While efforts ensure accuracy, results may not always reflect real-world scenarios. Verify results with other sources and consult professionals for critical applications. Contact us for any suggestions or corrections.