# Nusselt Number Calculator

Created by Arturo Barrantes
Reviewed by Steven Wooding
Based on research by
Holman JP Heat Transfer, 9th Edition January 1st, 2002See 1 more source
Incropera FP Foundation of Heat and Mass Transfer, 6th Edition (March 10th, 2006)
Last updated: Dec 16, 2022

The Nusselt number calculator helps you determine the relation between convective heat and conduction heat transfer between a fluid and a surface. This article will cover the Nusselt number equation, the acceptable range for the Nusselt number in natural convection, and when fluid flows through a pipe. Thus, if you have ever wondered how much convection can enhance heat transfer, this article is for you.

## What is the Nusselt number?

The Nusselt number is a dimensionless number that engineers use to describe the heat transfer rate between a fluid and a solid surface. It indicates how much heat transfer occurs due to convection compared to conduction.

Because of its meaning, engineers use the Nusselt number when designing equipment where heat transfer is the main objective, such as boilers or machinery cooling pipes.

Because of the primary definition of the Nusselt number, its equation is as follows:

$\footnotesize \text{Nu} = \frac{q_{\rm conv}}{q_{\rm cond}},$

where:

• $\rm Nu$Nussel number, dimensionless;
• $q_{\rm conv}$Heat transfer due to convection; and
• $q_{\rm cond}$Heat transfer due to conduction.

A higher Nusselt number indicates efficient convective heat transfer, which will increase the overall heat transfer coefficient, as we explain in the heat transfer coefficient calculator.

## How to calculate the Nusselt number?

We can write the Nusselt number formula explained above as a relation of convective heat transfer coefficient, characteristic length, and fluid thermal conductivity:

$\footnotesize \text{Nu} = \frac{h_{c} \times L}{k_{f}},$

where:

• $h_{c}$Convective heat transfer coefficient, expressed in $\footnotesize \rm{ W/m^2⋅K}$.
• $L$Characteristic length of the solid surface, expressed in $\footnotesize \rm{m}$. Remember that the characteristic length is typically the solid surface's length in the heat transfer direction.
• $k_{f}$Thermal conductivity of the fluid, expressed in $\footnotesize \rm{ W/m⋅K}$.

The above formula applies in all situations, but what if we cannot calculate the convective heat transfer coefficient? In that situation, we must use empirical correlations that involve other heat transfer dimensionless numbers such as the Reynolds number, Rayleigh number, and Prandtl number. Different combinations between them help you calculate the Nusselt number for natural convection and forced flow convection.

💡 For adding those extra fluid numbers, you can use the advanced mode of the Nusselt number calculator. Besides, if you are interested in learning more about forced fluid flow, check the Reynolds number calculator and the Prandtl number calculator.

## What is the Nusselt number for natural convection?

For natural convection, the Nusselt number range is over 1 and below 10, and the Rayleigh number determines it.

For vertical plates and cylinders like pipes, we have the following equation:

${\rm Nu} = C \times {\rm Ra}^{n}$

where:

• $\rm Ra$Rayleigh number; and,
• $C$, and $n$Nusselt number coefficients for natural convection.

We are going to cover laminar and turbulent scenarios for vertical plates and isothermic cylinders:

Flow

Usability

C coefficient

n coefficient

Vertical plate

Laminar

$\rm 10^4 \leq Ra \leq 10^9$

0.59

1/4

Turbulent

$\rm 10^9 \leq Ra \leq 10^{13}$

0.1

1/3

Horizontal Isothermic cylinder (pipe)

Laminar

$\rm 10^4 \leq Ra \leq 10^7$

0.48

1/4

Turbulent

$\rm 10^7 \leq Ra \leq 10^{12}$

0.125

1/3

## What is the Nusselt number for forced fluid convection?

Because we have flow velocity, we change the strategy and start using not only the Reynolds number but also the Prandtl number:

${\rm Nu} = C \times {\rm Re}^{m} \times {\rm Pr}^{n},$

where:

• $\rm Re$Reynolds number; and
• $\rm Pr$Prandtl number – both dimensionless.

We are going to cover laminar and turbulent scenarios for flat plates and isothermic cylinders/pipes:

*Const.: Constant.

Flow

Usability

C coefficient

m coefficient

n coefficient

Flat plate

Laminar

$\rm 0.6 < Pr$

0.664

0.5

0.33

Turbulent

$\rm 5 \times 10^5 < Re < 10^7$

0.037

0.8

0.33

$\rm 0.6 < Pr < 60$

Horizontal Isothermic cylinder (pipe)

Laminar

Const. surface temperature

Nu=3.66

Laminar

Const. surface heat flux

Nu=4.36

Turbulent

$\rm 10^4 < Re$

0.023

0.8

0.4 for heating

$\rm 0.6 < Pr < 160$

0.3 for cooling

Do not worry if you think we have so many formulas. Our excellent Nusselt number calculator includes them all in advanced mode.

## How to obtain the Nusselt number when checking the flow through pipe?

Let's assume you want to preserve the temperature of a natural gas pipe, and you will use a water heat exchanger. Consider the gas going through a pipe and getting heated with an isothermic external temperature. To obtain the Nusselt number of the fluid, we do the following:

1. Recognize it is a forced fluid/inside of a pipe/heating situation; thus, the Nusselt number equation we are going to use is $\rm \footnotesize 0.023 \times Re^{0.8} \times Pr^{0.4}$.

2. Assuming $\rm\footnotesize Re = 10.1^4$ and $\rm\footnotesize Pr = 1$, we obtain a Nusselt number of 37.64.

Therefore, the heating made by external water to the gas inside the pipe improves the heat transfer, and the heat transferred through convection is 37.6 times compared to the conductive heat transfer. This is a proper result of highly turbulent flow, which is true because we have a Reynolds number above 3500.

## FAQ

### What does a Nusselt number of 1 means?

A Nusselt number of 1 means no heat transfer enhancement due to convection. In other words, the heat transfer through a fluid layer occurs only by pure conduction. A Nusselt number greater than 1 indicates that the heat transfer is improved by convection. For a value between 1-10, we have the Nusselt number natural convection range.

### How to get the Nusselt number for a turbulent fluid?

We have to use the forced flow Nusselt number equation. Follow these steps:

1. Specify the geometry of the heat transfer surface.

2. Use:

• Nu = 0.037 × Re0.8 × Pr0.33 for flat plates.

• Nu = 0.023 × Re0.8 × Pr0.4 for pipes.

Above Re is the Reynolds number, and Pr is the Prandtl number.

### How to interpret a Nusselt number of 10?

A Nusselt number of 10 indicates that the heat transfer is enhanced ten times because of convection. In this situation, we can assume convection is being forced, thus increasing heat transfer because of higher mass flow than natural convection.

### What are some use cases for Nusselt number?

The Nusselt number helps determine the convective heat transfer coefficient by using flow properties such as speed & temperature and heat exchanger plates/pipes geometry. Engineers use the Nusselt number formula result for designing:

• Heat exchangers in boilers.
• Cooling systems.
• Any other system that involves convective heat transfer.
Arturo Barrantes
Characteristic length
m
Convection coefficient
W/(m²·K)
Fluid thermal conductivity
W/(m·K)
Nusselt number (Nu)
No
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