# Prandtl Number; Explanation, Formula and Calculator

## What is the Prandtl Number?

In heat transfer and fluid mechanics, we are using different kinds of coefficients and numbers. One of these coefficients is the Prandtl number. This is a very important coefficient that we need to know about it. Here we explained why we calculate it.

## What is Prandtl Number?

Firstly, to understand the Prandtl number, we need to understand the velocity and thermal boundary layers of fluids flowing on surfaces.

As you know from fluid mechanics, the velocity boundary layer is where the velocity of flowing fluid over a plate is zero. The thickness of this layer changes according to the different fluids.

### Thermal Boundary Layer

And there is a layer that we call as thermal boundary layer of flowing fluid over a plate. The surface of the plate has a temperature, and the flowing fluid has a temperature. There is a forced convection heat transfer between the surface of the layer and the fluid. Like the velocity layer, we have a thermal boundary layer in which the temperature of the fluid is equal to the surface temperature of the plate.

If we go through downstream, the thickness of the thermal boundary layer increases.

So, the importance of the Prandtl number emerges here. This is the ratio between the velocity boundary layer and the thermal boundary layer.

### Prandtl Number Formula

We calculate the Prandtl number with this formula;

In this formula;

- μ is the dynamic viscosity which has the unit of lb/(ft hr) or kg/m·s in SI units.
- Cp is the specific heat capacity of the flowing fluid under constant pressure. This value has the unit of kj/(kg.K) or Btu/lboF.
- k is the thermal conductivity which has the unit of W.m-1K-1or BTU.h-1.ft-1.℉-1 in English units.

Prandtl number is a dimensionless value.

According to this formula, with the increasing dynamic viscosity, it increases. And also, with the increasing specific heat capacity, this number increases. With the increasing thermal cınductivity, this number decreases.

### Why the Prandtl Number is Important?

For example, liquid metals have very good thermal conductivity values. They conduct heat very well. So, if the liquid metals flow on a surface, the thermal boundary layer will be very thick. And also, the boundary velocity layers of liquid metals are comparable with general fluids. So, the Prandtl Number of the metal fluids will be very low for them such as below 1.

Also for heavy oils, the velocity boundary layers will be very thick because o the dynamic viscosity. And the thermal boundary layers will be thin because they are not conducting the heat in a good way. So, the Prandtl number for heavy oils will be very high around 100.000.

### Forced Cooling Applications

According to the facts above, we prefer low Prandtl number liquids in forced cooling applications. For example, cooling of machinery that is working at high temperatures, and liquid metals are innovations. But the most important drawback of liquid metals, they give very bad corrosions on the surfaces that they are flowing. Because they are very heavy.

So, it is very important to select fluids that have low dynamic viscoısities and specific heat capacities and high thermal conductivities for forced cooling applications.

## Prandtl Number Calculator

You may see that the calculation of Prandtl numbers is very simple. But in fluid mechanics and other calculations, you will use Prandtl number calculations lots of the time. We prepared a calculator that you can use in these calculations. So, you will gain lots of time and effort by using this calculator in your complex calculations.

The use of this calculator is very simple. Firstly, you just need to enter the dynamic viscosity, the specific heat capacity of the fluid, and the thermal conductivity. And then, click on the ‘Calculate!’ button to see the Prandtl number of your fluid.

If you want to make further calculations, you just need to click on the ‘Reset’ button. And then re-enter the values.

### Examples of Prandtl Number Calculation

Check the example below to understand how to calculate the Prandtl number.

**Example:** We have a fluid that has a dynamic viscosity of *3*5 lb/(ft hr.). And the specific heat capacity of this fluid is 0.3 Btu/(lbm. °F). Also, the thermal conductivity is 0.06 Btu/(h.ft2 °F/ft). And then, calculate the it.

The calculation is very simple. So, if we put the given variables inside the formula;

Pr = 35×0.3/0.06 = 175.

So, this fluid has a quite high Prandtl number. It is not a good forced-cooling fluid.

## Conclusion

So, the Prandtl number is a very important value that shows an important feature of a fluid. Also, in convection heat transfer, it has a very important place. You just need to know the thermal conductivity, dynamic viscosity, and the specific heat capacity of the fluid to calculate it.

Finally, do not forget to leave your comments and questions below about this topic.

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NOTES:

Above all, Mechanicalland does not accept any responsibility for calculations made by users in calculators. A good engineer must check calculations again and again.

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## FAQs About Prandtl Number

**Does the Prandtl number change with temperature?**Yes. The temperature has a great influence on this number. Because, with the increasing temperature, specific heat capacity, thermal conductivity, and dynamic viscosities change.

**How to calculate the Prandtl number?**You can calculate this number by dividing the multiplication of dynamic viscosity and specific heat capacity with thermal conductivity.

**What does the Prandtl number tell us?**This number tells us the ratio between the velocity boundary layer and the thermal boundary layer of the flowing fluid over a surface.