# Prandtl Number Calculator

Created by Rahul Dhari
Reviewed by Steven Wooding
Last updated: Dec 13, 2021
Table of contents:

This Prandtl number calculator helps you determine this dimensionless number for different fluids. The Prandtl number is a function of the fluids' viscosity and thermal diffusivity. This dimensionless entity relates the momentum transport to the heat transport.

The value of the Prandtl number tells us by what means the heat would diffuse faster, i.e., whether heat conduction is more dominant than convection for a fluid. Prandtl number plays a vital role in heat transfer via fluids — a phenomenon extensively used in boundary layer flows. Read on to understand what the Prandtl number is and how to calculate the Prandtl number of air?

## What is Prandtl number?

Consider a fluid, say water flowing over a hot flat plate. If you want to find out the rate of energy or heat transferred via conduction or convection phenomenon, a dimensionless number known as the Prandtl number is used. In other words, the Prandtl number is the ratio of momentum diffusivity to thermal diffusivity.

$\footnotesize \text{Pr} = \frac{\text{Momentum transport}}{\text{Thermal (or heat) transport}} = \frac{\nu}{\alpha}$

Momentum transport (or diffusivity): In simple terms, it is the diffusion of momentum between particles in a fluid. The momentum diffusivity is also known as kinematic viscosity. We know that dynamic viscosity $\mu$ refers to the internal resistance of fluid layers against each other. The kinematic viscosity $\nu$ is the ratio of dynamic viscosity and the density of fluid, $\rho$. Mathematically, this is:

$\quad \nu = \frac{\mu}{\rho}$

The momentum diffusivity (or kinematic viscosity) have dimensions as area per time, i.e., $\text{m}^2/\text{s}$ or $\text{ft}^2/\text{s}$.

Thermal (or heat) transport (or diffusivity): It is the rate of heat transfer from one point to another. In other words, how fast can the heat travel from one location to another. The thermal diffusivity is measured in the units of area per time, i.e., $\text{m}^2/\text{s}$ or $\text{ft}^2/\text{s}$, and is a function of thermal conductivity specific heat capacity, and density of fluid. For a fluid having thermal conductivity, we can write $k$ and specific heat at constant pressure $C_p$, the thermal diffusivity, $\alpha$ as:

$\quad \alpha = \frac{k}{\rho C_p}$

An alternative Prandtl number formula can be written in terms of dynamic viscosity, $\mu$:

$\quad \text{Pr} = \frac{\mu C_p}{k}$

There are several variations of defining Prandtl number i.e., it is the ratio of:

• Kinematic viscosity to thermal diffusivity; or
• Momentum boundary layer to the thermal boundary layer.

Mathematically, we can write:

$\quad \text{Pr}^{-\frac{1}{3}} = \frac{\delta_t}{\delta}, 0.6 \le \text{Pr} \le 50$

where:

• $\delta_t$ – Thermal boundary layer thickness; and
• $\delta$ – Momentum boundary layer thickness.

## How to calculate Prandtl number?

To calculate Prandtl number:

1. Enter the dynamic viscosity, $\mu$ of the fluid.
2. Insert the specific heat capacity, $C_p$ of the fluid.
3. Fill in the thermal conductivity, $k$ for the fluid.
4. The calculator will return the Prandtl number, $\text{Pr}$.

Alternatively, you can utilize the preloaded data of common fluids from the list to directly obtain the Prandtl number of the said fluid.

💡 You can also use the advanced mode of the calculator in case you want to enter other parameters like kinematic viscosity, thermal diffusivity, and density of the fluid.

## Example: Using the Prandtl number calculator

Use the calculator to find the Prandtl number of water, given the following data:

Density, $\rho = 997 \text{ kg/m}^3$,
Dynamic viscosity, $\mu = 1.002 \text{ mPa⋅s}$,
Specific heat, $C_p = 4184 \text{ J/kg⋅K}$, and
Thermal conductivity, $k = 0.607 \text{ W/m⋅K}$.

To calculate Prandtl number of water:

1. Enter the dynamic viscosity, $\mu = 1.002 \text{ mPa⋅s}$.
2. Insert the specific heat capacity, $C_p = 4184 \text{ J/kg⋅K}$.
3. Fill in the thermal conductivity, $k = 0.607 \text{ W/m⋅K}$.
4. Using the Prandtl number formula:
$\scriptsize \text{Pr} = \frac{\mu C_p}{k} = \frac{1.002 × 10^{-3} * 4184}{0.607} = 6.90$

🔎 Use the advanced mode and enter the value of density to receive the values of momentum and thermal diffusivity.

## Prandtl number of different fluids

The following table has the Prandtl number for some common fluids:

Fluid

Prandtl number

Air

0.7-0.73

Water

0.69

Seawater

7.2-13.4

Oxygen

0.63

Argon

22.7

## Physical significance of Prandtl number

The physical significance of the Prandtl number is that when the number is less than 1, the conductive heat transfer is a more dominant phenomenon, i.e., the significant component of heat is transferred via conduction compared to convection. Similarly, when the Prandtl number is greater than 1, the convective heat transfer is more significant than conduction. For the fluids having a Prandtl number near 1, the heat transfer rate is more or less similar for both conduction and convection.

## FAQ

### What do you mean by Prandtl number?

Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. In other words, it is the ratio of heat transfer rate via convection and conduction. For the boundary layer phenomenon, it is known as the ratio of thicknesses of momentum and thermal boundary layers.

### What are the factors that affect Prandtl number of a fluid?

The factors affecting Prandtl number are:

• Dynamic viscosity (μ);
• Density;
• Thermal conductivity (k); and
• Specific heat capacity (Cp) of the fluid.

Mathematically, Pr = μ × Cp / k. The Prandtl number also varies with temperature.

### How do I calculate the Prandtl number?

To calculate the Prandtl number:

1. Multiply dynamic viscosity with the specific heat of the fluid.
2. Divide the product by the thermal conductivity of the fluid.

### What is the Prandtl number of water?

The Prandtl number of water is 6.9. For a dynamic viscosity of 1.002 mPa⋅s, thermal conductivity of 0.607 W/m⋅K and specific heat of 4184 J/kg⋅K, the Prandtl number is calculated as Pr = 0.001002 × 4184 / 0.607 = 6.9.

### What is the Prandtl number of air?

The Prandtl number of water is 0.715. For a dynamic viscosity of 0.0182 mPa⋅s, thermal conductivity of 0.025596 W/m⋅K and specific heat of 1006 J/kg⋅K, the Prandtl number is calculated as Pr = 0.0000182 × 1006 / 0.025596 = 0.715.

Rahul Dhari
Select the fluid from the list below to begin calculation of Prandtl number.
Fluid
Air
Properties of fluid
Dynamic viscosity (μ)
psi
•s
Specific heat (Cₚ)
BTU/(lb·°F)
Thermal conductivity (k)
W/(m⋅K)
Dimensionless number
Prandtl number (Pr)
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